When you dial a phone number on your iphone, how does the

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1 5 Trigonometric Identities When you dial a phone number on your iphone, how does the smart phone know which key you have pressed? Dual Tone Multi-Frequency (DTMF), also known as touch-tone dialing, was developed by Bell Labs in the 960s. The Touch-Tone system also introduced a standardized keypad layout. After testing 8 different layouts, Bell Labs eventually chose the one familiar to us today, with in the upper-left and 0 at the bottom between the star and the pound keys. The keypad is laid out in a 4 3 matrix, with each row representing a low frequency and each column representing a high frequency. Jason Brindel Commercial/Alamy FREQUENCY 09 HZ 336 HZ 477 HZ 697 Hz Hz Hz Hz * 0 # When you press the number 8, the phone sends a sinusoidal tone that combines a low-frequency tone of 85 hertz and a high-frequency tone of 336 hertz. The result can be found using sum-to-product trigonometric identities.

2 IN THIS CHAPTER, we will review basic identities and use those to simplify trigonometric expressions. We will verify trigonometric identities. Specific identities that will be discussed are sum and difference, double-angle and half-angle, and product-to-sum and sum-to-product. Music and touch-tone keypads are applications of trigonometric identities. The trigonometric identities have useful applications, and they are used most frequently in calculus. TRIGONOMETRIC IDENTITIES 5. Trigonometric Identities 5. Sum and Difference Identities 5.3 Double-Angle Identities 5.4 Half-Angle Identities 5.5 Product-to-Sum and Sum-to-Product Identities Verifying Trigonometric Identities Sum and Difference Identities for the Cosine Function Sum and Difference Identities for the Sine Function Sum and Difference Identities for the Tangent Function Applying Double-Angle Identities Applying Half-Angle Identities Product-to-Sum Identities Sum-to-Product Identities LEARNING OBJECTIVES Verify a trigonometric identity. Apply the sum and difference identities. Apply the double-angle identities. Apply the half-angle identities. Apply the product-to-sum and sum-to-product identities. 57

3 SECTION 5. TRIGONOMETRIC IDENTITIES SKILLS OBJECTIVES Simplify trigonometric expressions using basic identities. Verify trigonometric identities. CONCEPTUAL OBJECTIVES Understand that there is more than one way to verify an identity. Understand that identities must hold for all values in the domain of the functions that are related by the identities. Verifying Trigonometric Identities Basic Identities In Section.4, we discussed the basic (fundamental) trigonometric identities: reciprocal, quotient, and Pythagorean. In mathematics, an identity is an equation that is true for all values for which the expressions in the equation are defined. If an equation is true only for some values of the variable, it is a conditional equation. If an equation is true for no values of the variable, then it is contradiction. The following are identities (true for all x for which the expressions are defined): Study Tip (n )p, where n is an integer, is np equivalent to, where n is an odd integer. IDENTITY x 3x (x )(x ) tan x sin x cos x sin x cos x TRUE FOR THESE VALUES OF X All real numbers (n )p All real numbers except x, where n is an integer ax np, where n is an odd integerb All real numbers The following are conditional equations (true only for particular values of x): EQUATION x 3x 0 tan x 0 sin x cos x TRUE FOR THESE VALUES OF X x and x x np, where n is an integer (n )p x, where n is an integer ax np, where n is an odd integerb The following are contradictions (not true for any values of x): EQUATION x 5 x 7 sin x cos x 5 TRUE FOR THESE VALUES OF X none none 58

4 5. Trigonometric Identities 59 The following boxes summarize the identities that were discussed in Section.4: R ECIPROCAL IDENTITIES RECIPROCAL IDENTITIES csc x sin x sec x cos x cot x tan x x np x np x np DOMAIN RESTRICTIONS n is an integer n is an odd integer n is an integer Study Tip The reciprocal identity cot x is only valid if both tan x tanx and cotx are defined. x np includes both integer multiples of p and integer p multiples of. QUOTIENT IDENTITIES RECIPROCAL IDENTITIES tan x sin x cos x cot x cos x sin x cos x 0 or x sin x 0 or x np DOMAIN RESTRICTIONS (n )p n is an integer n is an integer PYTHAGOREAN IDENTITIES RECIPROCAL IDENTITIES sin x cos x tan x sec x cot x csc x DOMAIN RESTRICTIONS (n )p cos x 0 or x n is an integer sin x 0 or x np n is an integer In the previous chapters, we have discussed even and odd functions that have these respective properties: TYPE OF FUNCTION ALGEBRAIC IDENTITY GRAPH Even Odd f(x) f(x) f(x) f(x) Symmetry about the y-axis Symmetry about the origin We have already pointed out in previous chapters that the sine function is an odd function and the cosine function is an even function. Combining this knowledge with the reciprocal and quotient identities, we arrive at the even-odd identities. EVEN ODD IDENTITIES Odd: Even: sin(x) sin x csc(x) csc x tan(x) tan x cot(x) cot x cos(x) cos x sec(x) sec x

5 60 CHAPTER 5 Trigonometric Identities Simplifying Trigonometric Expressions Using Identities In Section.4, we used the basic trigonometric identities to find values for trigonometric functions, and we simplified trigonometric expressions using the identities. We now will use the basic identities and algebraic manipulation to simplify more complicated trigonometric expressions. In simplifying trigonometric expressions, one approach is to first convert all expressions into sines and cosines and then simplify. We will try that approach here. EXAMPLE Simplify the expression tanxsinx cosx. Simplifying Trigonometric Expressions Classroom Example 5.. Simplify sec x cos 3 x. Answer: sin x Write the tangent function in terms of the sine and sin x cosine functions, tan x. cos x Write as a fraction with a single quotient by finding a common denominator, cos x. Recognize the Pythagorean identity: sin x cos x. Use the reciprocal identity, sec x cos x. tan x sin x cos x tanx a sin x b sin x cos x cos x sin x cos x cos x sin x cos x sin x cos x cos x cos x sec x cos x cos x Answer: csc x YOUR TURN Simplify the expression cot x cos x sin x y x y = x + y = x x p In Example, tan x and sec x are not defined for odd integer multiples of. In the Your Turn, cot x and csc x are not defined for integer multiples of p. Both the original expression and the simplified form are governed by the same restrictions. There are times when the original expression is subject to more domain restrictions than the simplified form, and thus special attention must be given to note all domain restrictions. x For example, the algebraic expression has the domain restriction x x because that value for x makes the value of the denominator equal to zero. If we forget to state the domain restrictions, we might simplify the algebraic expressions x (x )(x ) x and assume this is true for all values of x. The correct x (x ) x result is x for x. In fact, if we were to graph both the original x expression y x and the line y x, they would coincide, except the graph of x the original expression would have a hole or discontinuity at x. In this chapter, it is assumed that the domain of the simplified expression is the same as the domain of the original expression.

6 5. Trigonometric Identities 6 EXAMPLE Simplifying Trigonometric Expressions Simplify the expression csc x sec x. Rewrite the expression in terms of quotients squared. csc x sec x a csc x b a sec x b Use the reciprocal identities to write the cosecant and secant functions in terms of sines and cosines: sin x csc x and cos x sec x. Recognize the Pythagorean identity: sin x cos x. sin x cos x Technology Tip Graph y csc x sec x and y. YOUR TURN Simplify the expression. cos x Answer: tan x Verifying Identities We will now use the trigonometric identities to verify, or prove, other trigonometric identities for all values for which the expressions in the equation are defined. For example, (sin x cos x) sin x cos x The good news is that we will know we are done when we get there, since we know the left side is supposed to equal the right side. But how do we get there? How do we verify that the identity is true? Remember that it must be true for all x, not just some x. Therefore, it is not enough to simply select values for x and show it is true for those specific values. Classroom Example 5.. csc u sec u Simplify. sec u csc u Answer: cot 4 u WORDS Start with one side of the equation (the more complicated side). Remember that (a b) a ab b and expand (sin x cos x). MATH (sin x cos x) sin x sin x cos x cos x Group the sin x and cos x terms and use the Pythagorean identity. sin x cos x (sin x cos x) sin x cos x When we arrive at the right side of the equation, then we have succeeded in verifying the identity. In verifying trigonometric identities, there is no one procedure that works for all identities. You can manipulate one side of the equation until it looks like the other side. Looking at where you want to be will help you make proper decisions on how to manipulate the one side. Here are two suggestions that are generally helpful:. Convert all trigonometric expressions to sines and cosines.. Write sums or differences of fractions (quotients) as a single fraction (quotient).

7 6 CHAPTER 5 Trigonometric Identities Study Tip Verifying identities: Start with one side of the equation and manipulate it until it looks like the other side of the equation. The following suggestions help guide the way to verifying trigonometric identities: G UIDELINES FOR VERIFYING TRIGONOMETRIC IDENTITIES Start with the more complicated side of the equation. Combine sums and differences of quotients into a single quotient. Use basic trigonometric identities. Use algebraic techniques to manipulate one side of the equation until it looks like the other side. Sometimes it is helpful to convert all trigonometric functions into sines and cosines. It is important to note that trigonometric identities must be valid for all values of the independent variable (usually x or u) for which the expressions in the equation are defined (domain of the equation). Classroom Example 5..3 Verify that (cos x )[ cos(x)] sin x. Answer: (cos x )[ cos(x)] (cos x )(cos x ) cos x sin x EXAMPLE 3 Verifying Trigonometric Identities Verify the identity ( sin x)[ sin(x)] cos x. Start with the more complicated side (left side). ( sin x)[ sin(x)] The sine function is odd: sin(x) sin x. ( sin x)( sinx) Eliminate the parentheses. sin x Apply the Pythagorean identity, sin x cos x. cos x EXAMPLE 4 Verify the identity Verifying Trigonometric Identities tan x cot x tan x cot x sin x cos x. Technology Tip tan x cot x Graph y and tan x cot x y sin x cos x. Start with the more complicated side of the equation. Use the quotient identities to write the tangent and cotangent functions in terms of the sine and cosine functions. Find the LCD of the numerator and the denominator. Multiply the numerator and the denominator by the sin x cos x. Distribute. sin x cos x a sin x cos x cos x sin x b sin x cos x a sin x cos x cos x sin x b tan x cot x tan x cot x sin x cos x LCD sin x cos x sin x cos x cos x sin x ± sin x cos x cos x sin x sin x cos x cos x sin x cos x cos x cos x sin x sin x cos x cos x sin x sin x cos x a sin x cos x b sin x cos x sin x sin x cos x sin x Recognize the Pythagorean identity in the denominator, sin x cos x. sin x cos x sin x cos x sin x cos x

8 5. Trigonometric Identities 63 EXAMPLE 5 Determining Whether a Trigonometric Equation Is an Identity Determine whether ( cos x)( cot x) 0 is an identity, conditional equation, or contradiction. Approach : Start with the left side of the equation and apply two Pythagorean identities. Apply the reciprocal identity, csc x sin x. Approach : Use the quotient identity to write the cotangent function in terms of the sine and cosine functions. Combine the expression in the second parentheses into a single fraction. Use the Pythagorean identity. ( cos x)( cot x) sin x csc x sin x csc x sin x ( cos x)( cot x) ( cos x)a cos x sin x b ( cos x)a sin x cos x b sin x ( cos x)a sin x cos x b sin x sin x sin x sin x sin x Classroom Example 5..5 Determine whether the following equation is an identity, conditional equation, or contradition: (sin x cos x). Answer: conditional Classroom Example 5..6 Verify that sec(x) cot(x). csc x Answer: sec(x) cot(x) csc x cos x cos x sin x sin x Since 0, this is not an identity, but rather a contradiction. EXAMPLE 6 sin(x) Verify that. cos(x) tan(x) Verifying Trigonometric Identities Technology Tip sin(x) Graph y cos(x) tan(x) and y. Start with the left side of the equation. Use the quotient identity to write the tangent function in terms of the sine and cosine functions. Simplify the product in the denominator. sin(x) cos(x) tan(x) sin(x) [cos(x)] c sin(x) cos(x) d sin(x) sin(x) Note: In the first step, we could have used the properties of even and odd functions: sin(x) sin x, cos(x) cos x, and tan(x) tan x.

9 64 CHAPTER 5 Trigonometric Identities Classroom Example 5..7 Prove that sec x[cot(x) tan(x)] csc(x). cos x Answer: sec x[cot(x) tan(x)] sec x(cot x tan x) cos x acos x sin x sin x cos x b x sin x cos x acos b sin x cos x csc x cos x sin x cos x csc(x) cos x EXAMPLE 7 Verifying a More Complicated Identity Verify that cot x sin x. csc x sin x Start with the left side of the equation. cot x csc x Use the Pythagorean identity, (csc x ) cot x csc x. csc x Distribute the negative throughout the numerator. csc x csc x ( csc x)( csc x) Factor the numerator (difference of two squares). csc x Divide out the csc x in both the numerator and denominator. csc x Use the reciprocal identity. sin x sin x Combine the two expressions into a single fraction. sin x SECTION 5. SUMMARY SMH In this section, we combined the basic trigonometric identities (reciprocal, quotient, Pythagorean, and even-odd) with algebraic techniques to simplify trigonometric expressions and verify more complex trigonometric identities. Two steps that are often used in both simplifying trigonometric expressions and verifying trigonometric identities are () writing all of the trigonometric functions in terms of the sine and cosine functions, and () combining sums or differences of fractions (quotients) into a single fraction (quotient). When verifying trigonometric identities, we work with the more complicated side (keeping the other side in mind as our goal). SECTION 5. EXERCISES SKILLS In Exercises 0, simplify each of the following trigonometric expressions.. sin x csc x. tan x cot x 3. sec(x) cot x sin x(cot x ) 6. cos x(tan x ) 7. (sin x cos x)(sin x cos x) 8. csc x sec x cos 4 x cot x tan x cos x tan x cot x cos x sin x tan x cot x cos x cos x sin x cot x cos x cos x sin(x) sin(x) 7. csc x csc x cos x cos x cos(x) cos(x) tan(x)cos(x) (sin x cos x) sin 4 x sin x tan x cot x tan x cot x cos x cot x csc x csc x

10 5. Trigonometric Identities 65 In Exercises 46, verify each of the trigonometric identities.. (sin x cos x) (sin x cos x). ( sin x)( sin x) cos x 3. (csc x )(csc x ) cot x 4. (sec x )(sec x ) tan x 5. tan x cot x csc x sec x 6. csc x sin x cot x cos x 7. sin x cos x sec x cos x 8. csc x sin x cos x sin x cot x tan x sec x csc x csc x sec x cos x cos x csc x sin x sin x sec x 33. sin x cos x cos x 34. sin x cos x sin x 35. sec x tan x 36. csc x cot x sec x tan x csc x cot x 37. csc x tan x sec x tan x 38. tan x sec x cot x cos x sin x sin x cos x csc x 39. cos x sin x sin x sin x cos x 40. sin x cos x 3 sin x cos x sin x 4. sec x(tan x cot x) csc x cos x 4. tan x(csc x sin x) cos x 43. cos (x) csc (x) sin x tan x 44. cos x( cos x) sin x cos x csc x 45. sin x sin x atan x b sec x 46. cos x tan x cos x sin x In Exercises 47 58, determine whether each equation is an identity, a conditional equation, or a contradiction. 47. cos x (tan x sec x)(tan x sec x) 48. cos x(tan x sec x)(tan x sec x) sin x 49. csc x cot x sec x tan x cot3 x 50. sin x cos x 0 5. sin x cos x 5. sin x cos x 53. tan x sec x 54. sec x tan x 55. sin x cos x 56. csc x cot x 57. sin x cos x 58. sin x cos x sin x cos x APPLICATIONS For Exercises 59 and 60, refer to the following: In calculus, when integrating expressions such as a x, a x, and x a, trigonometric functions are used as dummy functions to eliminate the radical. Once the integration is performed, the trigonometric function is un-substituted. The following trigonometric substitutions (and corresponding trigonometric identities) are used to simplify these types of expressions. EXPRESSION SUBSTITUTION TRIGONOMETRIC IDENTITY a x x asin u p u p sin u cos u a x x atan u p u p tan u sec u x a x asec u 0 u p or p u 3p sec u tan u When simplifying, it is important to remember that x if x 0 ƒ x ƒ e x if x 0

11 66 CHAPTER 5 Trigonometric Identities 59. Calculus (Trigonometric Substitution). Start with the expression a x and let x a sin u. Assuming p, simplify the original expression so that it u p contains no radicals. 60. Calculus (Trigonometric Substitution). Start with the expression a x and let x a tan u. Assuming p, simplify the original expression so that it u p contains no radicals. 6. Harmonic Motion. A weight is tied to the end of a spring and then set into motion. The displacement of the weight from equilibrium is given by the equation sin(t) cos(t) y 3c where t is time in seconds. sin(t) sec(t) d, Simplify the equation and then describe how long it takes for the weight to go from its minimum to its maximum displacement. 6. Harmonic Motion. A weight is tied to the end of a spring and then set into motion. The displacement of the weight from equilibrium is given by the equation sin at p b y 4 csc at p, b where t is time in seconds. Simplify the equation and then describe the maximum displacement with respect to the equilibrium position, and how long it takes for the weight to first achieve its maximum starting at t 0. CATCH THE MISTAKE In Exercises 63 66, explain the mistake that is made. 63. Verify the identity Start with the left side of the equation. Write the tangent and cotangent functions in terms of sines and cosines. Cancel the common cosine in the first term and sine in the second term. cos x sin x sin x cos x. tan x cot x cos x sin x cos x This is incorrect. What mistake was made? cos 3 x sec x 64. Verify the identity sin x. sin x Start with the left side of cos 3 x sec x the equation. sin x cos x sin x tan x cot x sin x cos x Rewrite secant in terms cos 3 x sin x of sine. sin x cos 3 x sin x sin x This is incorrect. What mistake was made? sin x cos x sin x 65. Determine whether the equation is a conditional equation tan x or an identity:. cot x Start with the left side of tan x the equation. cot x sin x Rewrite the tangent and cos x cotangent functions in terms of sines and cosines. cos x sin x sin x cos x tan x Let x p Note: tan a p b tan x Since, this equation is an identity. cot x This is incorrect. What mistake was made? 66. Determine whether the equation is a conditional equation or an identity: ƒ sin x ƒ cos x. Start with the left side of the equation. ƒ sin x ƒ cos x (n )p Let x, where n is an integer. (n )p (n )p ` sin a b` cos a b ƒ ƒ 0 Since ƒ sin x ƒ cos x, this is an identity. This is incorrect. What mistake was made?

12 { { { { 5. Trigonometric Identities 67 CONCEPTUAL In Exercises 67 70, determine whether each statement is true or false. 67. If an equation is true for some values (but not all values), then it is still an identity. 68. If an equation has an infinite number of solutions, then it is an identity. 69. The following is an identity true for all values in the domain of the functions: tan x sec x. 70. The following is an identity true for all values in the domain of the functions: csc x cot x. 7. In what quadrants is the equation cos u sin u true? 7. In what quadrants is the equation cos u sin u true? 73. Do you think sin(a B) sin A sin B? Why? 74. Do you think cos A AB cos A? Why? CHALLENGE 75. Simplify (a sin x b cos x) (b sin x a cos x). cot 3 x 76. Simplify cot x. cot x sin 4 x 77. Verify the trigonometric identity. cos x cos 4 x 78. Verify the trigonometric identity sin x sin x sin 3 x sin x. cos x a c d 79. Simplify the expression if a sin x, b b cos x, c cot x, and d csc x. a b 80. Simplify the expression if a sin x and ab b cos x. TECHNOLOGY In the next section you will learn the sum and difference identities. In Exercises 8 84, we illustrate these identities with graphing calculators. 8. Determine the correct sign ( or ) for cos(a B) cos A cos B sin A sin B by graphing? Y cos(a B), Y cos A cos B sin A sin B, and Y cos A cos B sin A sin B in the same viewing rectangle for chosen values for A and B, with A B. 8. Determine the correct sign ( or ) for cos(a B) cos A cos B sin A sin B by graphing? Y cos(a B), Y cos A cos B sin A sin B, and Y cos A cos B sin A sin B in the same viewing rectangle for chosen values for A and B with A B. 83. Determine the correct sign ( or ) for sin(a B) sin A cos B cos A sin B by graphing? Y sin(a B), Y sin A cos B cos A sin B, and Y sin A cos B cos A sin B in the same viewing rectangle for chosen values for A and B, with A B. 84. Determine the correct sign ( or ) for sin(a B) sin A cos B cos A sin B by graphing? Y sin(a B), Y sin A cos B cos A sin B, and Y sin A cos B cos A sin B in the same viewing rectangle for chosen values for A and B, with A B.

13 SECTION 5. SUM AND DIFFERENCE IDENTITIES SKILLS OBJECTIVES Find exact values of trigonometric functions for certain rational multiples of p by using sum and difference identities. Develop new identities from the sum and difference identities. CONCEPTUAL OBJECTIVE Understand that a trigonometric function of a sum is not equal to sum of the trigonometric functions. In this section, we will consider trigonometric functions with arguments that are sums and differences. In general, f (A B) f (A) f (B). First, it is important to note that function notation is not distributive: cos(a B) cos A cos B This principle is easy to prove. Let A p and B 0; then cos(a B) cos(p 0) cos p cos A cos B cos p cos 0 0 Since 0, we know that cos(a B) cos A cos B. In this section, we will derive some new and important identities (sum and difference identities for the cosine, sine, and tangent functions) and revisit cofunction identities. We begin with the familiar distance formula, from which we can derive the sum and difference identities for the cosine function. From there we can derive the sum and difference formulas for the sine and tangent functions. Distance Formula Sum and Difference Identities for Cosine Cofunction Identities Sum and Difference Identities for Sine Sum and Difference Identities for Tangent Before we start deriving and working with trigonometric sum and difference identities, let us first discuss why these are important. Sum and difference identities (and later product-to-sum and sum-to-product identities) are important because they allow the calculation of trigonometric function values in functional (analytic) form and often lead to evaluating expressions exactly (as opposed to approximating them with calculators). The identities developed in this chapter are useful in such applications as musical sound where they allow the determination of the beat frequency. In calculus these identities will simplify the integration and differentiation processes. Sum and Difference Identities for the Cosine Function 68 Derivation of the Sum and Difference Identities for the Cosine Function Recall from Section 3.4 that the unit circle approach for defining trigonometric functions gave the relationship between the coordinates along the unit circle and the sine and cosine functions. Specifically, the x-coordinate corresponds to the value of the cosine function,

14 5. Sum and Difference Identities 69 and the y-coordinate corresponds to the value of the sine function for a given angle u and the point (x, y) where the terminal side of the angle u intersects the unit circle. y (, 0) (0, ) r = (cos, sin) (, 0) x (0, ) Let us now draw the unit circle with two angles, A and B, realizing that the two terminal sides of these angles form a third angle, A B. (0, ) P P (, 0) y (, 0) x (0, ) If we label the points P (cos A, sin A) and P (cos B, sin B), we can then draw a segment P P connecting points and P. P y (0, ) P P = (cos, sin ) (, 0) P P = (cos, sin ) x (, 0) (0, ) If we rotate the angle clockwise so the central angle A B is in standard position, then the two points where the initial and terminal sides intersect the unit circle are P 4 (, 0) and P 3 (cos(a B), sin(a B)), respectively. y P 3 = (cos( ), sin( )) (0, ) (, 0) x P 4 = (, 0) (0, )

15 70 CHAPTER 5 Trigonometric Identities Study Tip The distance from the point (x, y ) to the point (x, y ) is given by the distance formula d (x x ) (y y ). The distance from P to P is equal to the length of the segment. Similarly, the distance from P 3 to P 4 is equal to the length of the segment. Since the lengths of the segments are equal, we say that the distances are equal. WORDS Set the distances (segment lengths) equal by applying the distance formulas. MATH (x (x 4 x 3 ) (y 4 y 3 ) x ) (y y ) Substitute (x, y ) (cos A, sin A) and (x, y ) (cos B, sin B) into the left side of the equation and (x 3, y 3 ) (cos(a B), sin(a B)) and (x 4, y 4 ) (, 0) into the right side of the equation. Square both sides of the equation. Eliminate the brackets. Regroup terms on each side and use the Pythagorean identity. Subtract from both sides. Divide by. We call the resulting identity the difference identity for cosine. (cos b cos a) (sin b sin a) ( cos(a b)) (0 sin(a b)) (cos b cos a) (sin b sin a) ( cos(a b)) (0 sin(a b)) cos b cos a cos b cos a sin b sin a sin b sin a cos(a b) cos (a b) sin (a b) cos a sin a cos a cos b sin a sin b cos b sin b cos a cos b sin a sin b cos(a b) cos a cos b sin a sin b cos(a b) cos a cos b sin a sin b cos(a b) cos(a b) cos (a b) sin (a b) cos(a b) cos a cos b sin a sin b We can now derive the sum identity for the cosine function from the difference identity for the cosine function and the properties of even and odd functions. WORDS Replace b with b in the difference identity. Simplify the left side and use properties of even and odd functions on the right side. We call the resulting identity the sum identity for cosine. MATH cos(a (b)) cos a cos(b) sin a sin(b) cos(a b) cos a(cos b) sin a(sin b) cos(a b) cos a cos b sin a sin b

16 5. Sum and Difference Identities 7 SUM AND DIFFERENCE IDENTITIES FOR THE COSINE FUNCTION Sum: Difference: EXAMPLE cos(a B) cos A cos B sin A sin B cos(a B) cos A cos B sin A sin B Finding Exact Values for the Cosine Function Evaluate each of the following cosine expressions exactly: a. cos 5 b. cos a 7p b Solution (a): Write 5 as a difference of known special angles. cos 5 cos(45 30 ) Write the difference identity for the cosine function. cos(a B) cos A cos B sin A sin B Substitute A 45 and B 30. Evaluate the expressions on the right exactly. cos 5 cos 45 cos 30 sin 45 sin 30 cos 5 a b a3 b a b a b Classroom Example 5.. Compute: a. cosa 7p b b. cos(5 ) Answer: 6 a. 4 6 b. 4 Technology Tip a. Use a TI calculator to check the values for cos a 7p and b 6. 4 Solution (b): 7p Write as a sum of known special angles. Write the sum identity for the cosine function. Substitute A p and B p 3 4. Evaluate the expressions on the right exactly. cos cos a 7p 3p b cos a4p b cos a 7p b cos ap 3 p 4 b cos(a B) cos A cos B sin A sin B cos a 7p b cos ap 3 b cos ap 4 b sin ap 3 b sin ap 4 b cos a 7p b a b a 3 b a b a b b. Use a TI calculator to check the 6 values of cos 5 and. 4 Be sure the calculator is set in degree mode. cos a 7p 6 b 4 7p Note: can also be represented as 05. YOUR TURN Use the sum or difference identities for the cosine function to evaluate each cosine expression exactly. a. cos 75 b. cos a 5p b Example illustrates an important characteristic of the sum and difference identities: that we can now find the exact trigonometric function value of angles that are multiples of 5 aor equivalently p. b Answer: a. b

17 7 CHAPTER 5 Trigonometric Identities Technology Tip a. Graph y sin(5x) sin( x) cos(5x) cos(x) and y cos(3 x). EXAMPLE Writing a Sum or Difference as a Single Cosine Expression Use the sum or the difference identity for the cosine function to write the expressions as a single cosine expression. a. sin(5x) sin( x) cos(5x) cos( x) b. cos x cos(3x) sin x sin(3x) Solution (a): Because of the positive sign, this will be a cosine of a difference. Reverse the expression and write the formula. cos A cos B sin A sin B cos(a B) Identify A and B. A 5x and B x Substitute A 5x and B x into the difference identity. cos(5x) cos(x) sin(5x) sin(x) cos(5x x) b. Graph y cos x cos(3x) sin x sin(3x) and y cos(4x). cos(5x) cos(x) sin(5x) sin(x) cos(3x) Notice that if we had selected A x and B 5x instead, the result would have been cos(3x), but since the cosine function is an even function, this would have simplified to cos(3x). Solution (b): Because of the negative sign, this will be a cosine of a sum. Reverse the expression and write the formula. cos A cos B sin A sin B cos(a B) Identify A and B. A x and B 3x Substitute A x and B 3x into the sum identity. cos x cos(3x) sin x sin(3x) cos(x 3x) cos x cos(3x) sin x sin(3x) cos(4x) Answer: cos(3x) YOUR TURN Write cos(4x) cos(7x) sin(4x) sin(7x) as a single cosine expression. Classroom Example 5.. Write as a single cosine. a. sin(px) sin a x b cos a x b cos(px) Cofunction Identities In Section.3, we discussed cofunction relationships for acute angles. Recall that a trigonometric function value of an angle is equal to the corresponding cofunction value of its complementary angle. Now we use the sum and difference identities for the cosine function to develop the cofunction identities for any angle u. b. sin(px) sin a x b cos a x b cos(px) Answer: a. cos cap b x d b. cos cap b xd WORDS Write the difference identity for the cosine function. Let A p and B u. Evaluate known values for the sine and cosine functions. MATH cos(a B) cos A cos B sin A sin B cos a p ub cos ap b cos u sin ap b sin u cos a p ub 0 cos u sin u cos a p ub sin u

18 5. Sum and Difference Identities 73 Similarly, to determine the other corresponding cofunction identity: WORDS Write the difference identity for the cosine function. MATH cos(a B) cos A cos B sin A sin B Let A p and B p u. cos c p ap ubd cos ap b cos ap ub sin ap b sin ap ub Evaluate the known values for the sine and cosine functions. cos u 0 cos a p ub sin ap ub cos u sin a p ub COFUNCTION IDENTITIES FOR THE SINE AND COSINE FUNCTIONS cosap ub sin u sin a p ub cos u Sum and Difference Identities for the Sine Function We can now use the cofunction identities for the sine and cosine functions together with the sum and difference identities for the cosine function to develop the sum and difference identities for the sine function. WORDS MATH Start with the cofunction identity. sin u cos a p ub Let u A B. sin(a B) cos c p (A B) d Regroup the terms in the cosine expression. sin(a B) cos ca p Ab B d Use the difference identity for the cosine function. sin(a B) cos a p Ab cos B sin ap Ab sin B Use the cofunction identities. sin(a B) cos a p Ab cos B sin ap Ab sin B sin A cos A sin(a B) sin A cos B cos A sin B Now we can derive the difference identity for the sine function using the sum identity for the sine function and the properties of even and odd functions.

19 74 CHAPTER 5 Trigonometric Identities WORDS Replace B with B in the sum identity. Simplify using even and odd identities. MATH sin[a (B)] sin A cos(b) cos A sin(b) sin(a B) sin A cos B cos A sin B SUM AND DIFFERENCE IDENTITIES FOR THE SINE FUNCTION Sum: Difference: sin(a B) sin A cos B cos A sin B sin(a B) sin A cos B cos A sin B Classroom Example 5..3 Compute: a. sin a 7p b b. sin (5 ) Answer: 6 a. a b 4 6 b. 4 EXAMPLE 3 Finding Exact Values for the Sine Function Use the sum or the difference identity for the sine function to evaluate each sine expression exactly. a. sin 75 b. sin a 5p b Solution (a): Write 75 as a sum of known special angles. sin 75 sin(45 30 ) Write the sum identity for the sine function. Substitute A 45 and B 30. sin(a B) sin A cos B cos A sin B sin 75 sin 45 cos 30 cos 45 sin 30 sin 75 a Evaluate the expressions on the right exactly. b a 3 b a b a b Answer: a. b Solution (b): 5p Write as a sum of known special angles. Write the sum identity for the sine function. Substitute A p and B p 6 4. Evaluate the expressions on the right exactly. Note: 5p can also be represented as 75. YOUR TURN Use the sum or the difference identity for the sine function to evaluate the sine expressions exactly. a. sin 5 b. sin a 7p b sin sin a 5p 3p b sin ap b sin a 5p b sin ap 6 p 4 b sin(a B) sin A cos B cos A sin B sin a 5p b sin ap 6 b cos ap 4 b cos ap 6 b sin ap 4 b sin a 5p b a b a 3 b a b a b sin a 5p 6 b 4

20 5. Sum and Difference Identities 75 We see in Example 3 that the sum and difference identities allow us to calculate exact values for trigonometric functions of angles that are multiples of 5 aor equivalently p as we saw with the cosine function. b, EXAMPLE 4 Writing a Sum or Difference as a Single Sine Expression Graph y 3 sin x cos(3x) 3 cos x sin(3x). Use the sum identity for the sine function to write the expression as a single sine expression. Factor out the common 3. Write the sum identity for the sine function. Identify A and B. Substitute A x and B 3x into the sum identity. Graph y 3 sin(4x). y 3[sin x cos(3x) cos x sin(3x)] sin A cos B cos A sin B sin(a B) A x and y 3[sin x cos(3x) cos x sin(3x)] B 3x sin(x 3x) sin(4x) 3 y x Technology Tip Graph y 3 sin x cos(3x) 3 cos x sin(3x) and y 3 sin(4x). 3 Sum and Difference Identities for the Tangent Function We now develop the sum and difference identities for the tangent function. WORDS Start with the quotient identity. Let x A B. Use the sum identities for the sine and cosine functions. To be able to write the right-hand side in terms of tangents, we multiply the numerator and the denominator by. cos A cos B MATH tan x sin x cos x sin(a B) tan(a B) cos(a B) sin A cos B cos A sin B tan(a B) cos A cos B sin A sin B sin A cos B cos A sin B cos A cos B tan(a B) cos A cos B sin A sin B cos A cos B

21 76 CHAPTER 5 Trigonometric Identities Classroom Example 5..4* Graph y sin(px) cos ap 3 xb. cos(px) sin ap xb 4 3 Answer: First, reduce this to y. sin ap xb 4 3 Divide out (cancel) common factors. tan(a B) sin A cos B cos A cos B cos A sin B cos A cos B cos A cos B sin A sin B cos A cos B cos A cos B a sin A sin B b a cos A cos B b a sin A cos A b a sin B cos B b Write the expressions inside the parentheses in terms of the tangent function. Replace B with B. Since the tangent function is an odd function, tan(b) tan B. tan(a B) tan[a (B)] tan(a B) tan(a B) tan A tan B tan A tan B tan A tan(b) tan A tan(b) tan A tan B tan A tan B SUM AND DIFFERENCE IDENTITIES FOR THE TANGENT FUNCTION Sum: tan A tan B tan(a B) tan A tan B Difference: tan A tan B tan(a B) tan A tan B EXAMPLE 5 Finding Exact Values for the Tangent Function Find the exact value of tan(a b) if sin a and cos b 3 4, given that the terminal side of a lies in quadrant III and the terminal side of b lies in quadrant II. STEP STEP Write the sum identity for the tangent function. Find tan a. The terminal side of a lies in quadrant III. tan a tan b tan(a b) tan a tan b y sin a y r 3 x 3 (x, ) x

22 5. Sum and Difference Identities 77 Solve for x. ( x y r ) Take the negative sign since we are in quadrant III. Find tan a. x 3 x 8 x tan a y x 4 Classroom Example 5..5 Find the exact value of cot(a b) if sin a 3, cos b 4, the terminal side of a is in quadrant III, and the terminal side of b is in quadrant II. STEP 3 Find tan b. The terminal side of b lies in quadrant II. (, y) y 4 y x Answer: Solve for y. () y 4 ( x y r ) y 5 STEP 4 Take the positive sign since we are in quadrant II. Find tan b. Substitute tan a 4 and tan b 5 into the sum identity for the tangent function. tan b y x tan(a b) y a 4 b A5B Multiply the numerator and the denominator by 4. 4 tan(a b) 4 a 4 5b a 30 4 b The expression tan(a b) rationalize the denominator can be simplified further if we It is important to note in Example 5 that right triangles have been superimposed in the Cartesian plane. The coordinate pair (x, y) can have positive or negative values, but the radius r is always positive. When right triangles are superimposed with one vertex at the point (x, y) and another vertex at the origin, it is important to understand that triangles have positive side lengths.

23 78 CHAPTER 5 Trigonometric Identities SECTION 5. SUMMARY SMH In this section, we derived the sum and difference identities for the cosine function using the distance formula. We then used these identities to derive the cofunction identities. The cofunction identities and sum and difference identities for the cosine function were used to derive the sum and difference identities for the sine function. The sine and cosine sum and difference identities were combined to determine the tangent sum and difference identities. The sum and difference identities enabled us to evaluate trigonometric expressions exactly for arguments that are integer multiples of 5 ai.e., p. b cos(a B) cos A cos B sin A sin B cos(a B) cos A cos B sin A sin B sin(a B) sin A cos B cos A sin B sin(a B) sin A cos B cos A sin B tan(a B) tan(a B) tan A tan B tan A tan B tan A tan B tan A tan B SECTION 5. EXERCISES SKILLS In Exercises 0, find exact values for each trigonometric expression.. sin a p. cos a p 3. cos a5p 4. sin a5p 5. sin a7p 6. cos a 7p b b b b b b 7. tan a p 8. tan a 3p 9. sin cos 95. tan(05 ). tan 65 b b 3. cot a p b 4. cot a5p b 5. sec ap b 6. sec a3p b 7. csc a7p b sec(95 ) 0. csc 85 csc a 3p b In Exercises 34, write each expression as a single trigonometric function.. sin(x) sin(3x) cos(x) cos(3x). sin x sin(x) cos x cos(x) 3. sin x cos(x) cos x sin(x) 4. sin(x) cos(3x) cos(x) sin(3x) 5. (sin A sin B) (cos A cos B) 6. (sin A sin B) (cos A cos B) 7. cos A xb sin A 5 xb cos A 5 xb sin A xb 8. cos 50 cos x sin 50 sin x 9. (sin A cos B) (cos A sin B) 30. (sin A cos B) (cos A sin B) 3. tan 49 tan 3 tan 49 tan 3 tan(p/8) tan(3p/8) tan(7p/) tan(p/6) tan 49 tan 3 tan 49 tan 3 tan(p/8) tan(3p/8) tan(7p/) tan(p/6) In Exercises 35 40, find the exact value of the indicated expression using the given information and identities. 35. Find the exact value of cos(a b) if cos a and cos b 3 4, if the terminal side of a lies in quadrant III and the terminal side of b lies in quadrant II. 36. Find the exact value of cos(a b) if cos a and cos b 3 4, if the terminal side of a lies in quadrant IV and the terminal side of b lies in quadrant II. 37. Find the exact value of sin(a b) if sin a 3 and sin b 5 5, if the terminal side of a lies in quadrant III and the terminal side of b lies in quadrant I.

24 5. Sum and Difference Identities Find the exact value of sin(a b) if sin a 3 and sin b 5 5, if the terminal side of a lies in quadrant III and the terminal side of b lies in quadrant II. 39. Find the exact value of tan(a b) if sin a 3 and cos b 5 4, if the terminal side of a lies in quadrant III and the terminal side of b lies in quadrant II. 40. Find the exact value of tan(a b) if sin a 3 and cos b 5 4, if the terminal side of a lies in quadrant III and the terminal side of b lies in quadrant II. In Exercises 4 50, determine whether each equation is a conditional equation or an identity. 4. sin(a B) sin(a B) sin A cos B sin ax p b cos ax p b sin(x) sin x cos x sin(a B) sin A sin B tan(p B) tan B 50. cos(a B) cos(a B) cos A cos B sin ax p b cos ax p b cos(x) cos x sin x cos(a B) cos A cos B tan(a p) tan A In Exercises 5 56, graph each of the functions by first rewriting it as a sine, cosine, or tangent of a difference or sum. 5. y cos a p 3 b sin x cos x sin ap 3 b y sin x sin ap 4 b cos x cos ap 4 b 54. p y cos a 3 b sin x cos x sin ap 3 b y sin x sin a p 4 b cos x cos ap 4 b 55. y sin x cos(3x) cos x sin(3x) 56. y sin x sin(3x) cos x cos(3x) APPLICATIONS For Exercises 57 and 58, refer to the following: f (x h) f (x) The difference quotient, is used to h approximate the rate of change of the function f and will be used frequently in calculus. 57. Difference Quotient. Show that the difference quotient for is cos x a sin h cos h f (x) sin x b sin x a b. h h 58. Difference Quotient. Show that the difference quotient for is sin x a sin h cos h f (x) cos x b cos x a b. h h For Exercises 59 and 60, refer to the following: A nonvertical line makes an angle with the x-axis. In the figure, we see that the line L makes an acute angle with the x-axis. Similarly, the line L makes an acute angle with the x-axis. In Exercises 59 and 60, use the following: tan u slope of L m tan u slope of L m y L L x 59. Relating Tangent and Scope. Show that tan(u u ) m m m m. 60. Relating Tangent and Scope. Show that tan(u u ) m m m m. For Exercises 6 and 6, refer to the following: An electric field E of a wave with constant amplitude A propagating a distance z is given by E A cos(kz ct) where k is the propagation wave number, which is related to the wavelength l by k p, c meters per second is l the speed of light in a vacuum, and t is time in seconds. 6. Electromagnetic Wave Propagation. Use the cosine difference identity to express the electric field in terms of both sine and cosine functions. When the quotient of the propagation distance z and the wavelength l are equal to an integer, what do you notice? 6. Electromagnetic Wave Propagation. Use the cosine difference identity to express the electric field in terms of both sine and cosine functions. When t 0, what do you notice?

25 80 CHAPTER 5 Trigonometric Identities 63. Functions. Consider a 5-foot ladder placed against a wall such that the distance from the top of the ladder to the floor is h feet and the angle between the floor and the ladder is u. a. Write the height h as a function of angle u. b. If the ladder is pushed toward the wall, increasing the angle u by 0, write a new function for the height as a function of u 0 and then express in terms of sines and cosines of u and Functions. Consider a 5-foot ladder placed against a wall such that the distance from the bottom of the ladder to the wall is x feet and the angle between the floor and the ladder is u. a. Write the distance x as a function of angle u. b. If the ladder is pushed toward the wall, increasing the angle u by 0, write a new function for the height as a function of u 0 and then express in terms of sines and cosines of u and Biology. By analyzing available empirical data, it has been determined that the body temperature of a species fluctuates according to the model T(t) 38.5 cos c p (t 3) d, 0 t 4 6 where T represents temperature in degrees Celsius and t represents time (in hours) measured from :00 P.M. (noon). Use an identity to express T(t) in terms of the sine function. 66. Health/Medicine. During the course of treatment of an illness, the concentration of a drug (in micrograms per milliliter) in the bloodstream fluctuates during the dosing period of 8 hours according to the model C(t) sin a p 4 t p b, 0 t 8 Use an identity to express the concentration C(t) in terms of the cosine function. Note: This model does not apply to the first dose of the medication as there will be no medication in the bloodstream. CATCH THE MISTAKE In Exercises 67 and 68, explain the mistake that is made. 67. Find the exact value of 5p Write as a sum. Distribute. tan a 5p b. tan a 5p b tan ap 4 p 6 b tan a p 4 b tan ap 6 b Evaluate the tangent p p function for and This is incorrect. What mistake was made? CONCEPTUAL 68. Find the exact value of tan a 7p 6 b. The tangent function is an even function. 7p Write as a sum. 6 Use the tangent sum identity, tan A tan B tan(a B) tan A tan B. Evaluate the tangent functions on the right. tan a 7p 6 b tan a7p 6 b 3 3 This is incorrect. What mistake was made? tan ap p 6 b tan p tan a p 6 b tan p tan a p 6 b In Exercises 69 7, determine whether each statement is true or false. 69. cos 5 cos 45 cos Simplify the expression cos(ax) cos(bx) sin(ax)sin(bx) sin a p b sin ap 3 b sin ap 6 b sin(90 x) cos x to a single trigonometric function. 74. Simplify the expression cos(ax) sin(bx) sin(ax) cos(bx) to a single trigonometric function. 7. cos(90 x) sin x

26 5.3 Double-Angle Identities 8 CHALLENGE 75. Verify that sin(a B C) sin A cos B cos C cos A sin B cos C cos A cos B sin C sin A sin B sin C. 76. Verify that cos(a B C) cos A cos B cos C sin A sin B cos C sin A cos B sin C cos A sin B sin C. 77. Although, in general, the statement sin(a B) sin A sin B is not true, it is true for some values. Determine some values of A and B that make this statement true. 78. Although, in general, the statement sin(a B) sin A sin B is not true, it is true for some values. Determine some values of A and B that make this statement true. (sin x cos y) 79. Verify the identity cot x tan y sin x cos y sin(x y) sin x cos y. sin x cos y cos y sin x 80. Verify the identity tan a cot b cos(a b) sec a csc b. TECHNOLOGY 8. In Exercise 57, you showed that the difference quotient for is cos x a sin h cos h f (x) sin x b sin x a b. h h Plot Y cos x a sin h h cos h b sin x a b h a. h b. h 0. c. h 0.0 What function does the difference quotient for f (x) sin x resemble when h approaches zero? for 8. In Exercise 58, you showed that the difference quotient for is sin x a sin h cos h f (x) cos x b cos x a b. h h Plot Y sin x a sin h h cos h b cos x a b h a. h b. h 0. c. h 0.0 What function does the difference quotient for f (x) cos x resemble when h approaches zero? for SECTION 5.3 DOUBLE-ANGLE IDENTITIES SKILLS OBJECTIVES Use the double-angle identities to find certain exact values of trigonometric functions. Use the double-angle identities to help in verifying identities. CONCEPTUAL OBJECTIVE Understand that the double-angle identities are derived from the sum identities. Applying Double-Angle Identities Throughout this text, much attention has been given to distinguishing between evaluating trigonometric functions exactly (for special angles) or approximating values of trigonometric functions with a calculator. In previous chapters, we could only evaluate trigonometric p functions exactly for reference angles of 30, 45, and 60 or, and note that 6, p 4, and p 3 p as of the previous section, we can now include multiples of among these special angles. Now we can use double-angle identities to evaluate trigonometric function values for other angles that are even integer multiples of the special angles or to verify other trigonometric identities. One important distinction now is that we will be able to find exact values of many functions using the double-angle identities without needing to know the actual value of the angle.

27 8 CHAPTER 5 Trigonometric Identities Derivation of Double-Angle Identities To derive the double-angle identities, we let A B in the sum identities: WORDS Write the identity for the sine of a sum. Let B A. Write the identity for the cosine of a sum. Let B A. MATH sin(a B) sin A cos B cos A sin B sin(a A) sin A cos A cos A sin A sin(a) sin A cos A cos(a B) cos A cos B sin A sin B cos(a A) cos A cos A sin A sin A cos(a) cos A sin A The double-angle identity for the cosine function can be written two other ways if we use the Pythagorean identity: WORDS. Write the identity for the cosine function of a double angle. Use the Pythagorean identity for cosine.. Write the identity for the cosine function of a double angle. Use the Pythagorean identity for the sine function. MATH cos(a) cos A sin A cos(a) cos A sin A sin A cos(a) sin A cos(a) cos A sin A cos A cos(a) cos A ( cos A) cos A cos A cos(a) cos A The tangent function can always be written as a quotient, tan(a) sin(a), if sin(a) cos(a) and cos (A) are known. Here we write the double-angle identity for the tangent function in terms of only the tangent function. WORDS Write the identity for the tangent of a sum. Let B A. MATH tan(a B) tan(a A) tan A tan B tan A tan B tan A tan A tan A tan A tan(a) tan A tan A

28 5.3 Double-Angle Identities 83 DOUBLE-ANGLE IDENTITIES FOR THE SINE, COSINE, AND TANGENT FUNCTIONS SINE COSINE TANGENT sin(a) sin A cos A cos(a) cos A sin A tan(a) tan A tan A cos(a) sin A cos(a) cos A Applying Double-Angle Identities EXAMPLE If cos x 3, find sin(x) given sin x 0. Finding Exact Values of Trigonometric Functions Using Double-Angle Identities Find sin x. Use the Pythagorean identity. Substitute cos x 3. sin x cos x sin x a 3 b y Solve for sin x, which is negative. sin x B 4 9 x x Find sin(x). Use the double-angle formula for the sine function. Substitute sin x 5 and cos x sin x B sin(x) sin x cos x sin(x) a 5 3 b a 3 b sin(x) 45 9 Classroom Example 5.3. If sin x 5 and cos x 0, find sin(x). Answer: YOUR TURN If cos x 3, find sin(x) given sin x 0. Answer: sin(x) 4 9

29 84 CHAPTER 5 Trigonometric Identities EXAMPLE Finding Exact Values Using Double-Angle Identities If sin x 4 5 and cos x 0, find sin(x), cos(x), and tan(x). y x 3 x 4 5 ( 3, 4) Solve for cos x. Use the Pythagorean identity. Substitute sin x 4 5. Solve for cos x, which is negative. Find sin(x). Use the double-angle identity for the sine function. Substitute sin x 4 5 and cos x 3 5. sin x cos x 4 a 5 b cos x cos x 9 5 cos x B sin(x) sin x cos x sin(x) a 4 5 b a3 5 b Find cos(x). Use the double-angle identity for the cosine function. Substitute sin x 4 and cos x sin(x) 4 5 cos(x) cos x sin x cos(x) a 3 5 b a 4 5 b Classroom Example 5.3. If sin x 5 and cos x 0, find cos(x) and cot(x). Answer: cos(x) 3 5 cot(x) 36 4 Answer: sin(x) 4 5 cos(x) 7 5 tan(x) 4 7 Find tan(x). Use the quotient identity. Let u x. Substitute sin(x) 4 and cos(x) cos(x) 7 5 tan u sin u cos u tan(x) sin(x) cos(x) tan(x) 4/5 7/5 tan(x) 4 7 Note: tan(x) could also have been found by first finding tan x sin x and then using cos x the value for tan x in the double-angle identity, tan(a) tan A. tan A YOUR TURN If cos x 3 5 and sin x 0, find sin(x), cos(x), and tan(x).

30 5.3 Double-Angle Identities 85 EXAMPLE 3 Verifying Trigonometric Identities Using Double-Angle Identities Verify the identity (sin x cos x) sin( x). Start with the left side of the equation. (sin x cos x) Technology Tip Graph y (sin x cos x) and y sin(x). Expand by squaring. Group the sin x and cos x terms. Recognize the Pythagorean identity. Recognize the sine double-angle identity. sin x sin x cos x cos x sin x cos x sin x cos x sin x cos x sin x cos x sin x cos x sin(x) (sin x cos x) sin(x) Classroom Example Prove that [cos(x) sin(x)] sin(4x). Answer: [cos(x) sin(x)] cos (x) sin(x) cos(x) sin (x) sin(x) cos(x) sin(4x) EXAMPLE 4 Verifying Multiple-Angle Identities Using Double-Angle Identities Verify the identity cos(3x) ( 4 sin x) cos x. Write the cosine of a sum identity. Let A x and B x. Recognize the double-angle identities. cos(a B) cos A cos B sin A sin B cos( x x) cos( x) cos x sin(x) sin x cos(3x) cos(x) cos x sin(x) sin x sin x sinxcosx cos(3x) cos x sin x cos x sin x cos x cos(3x) cos x 4 sin x cos x Factor out the common cosine term. cos(3x) ( 4 sin x) cos x

31 86 CHAPTER 5 Trigonometric Identities Technology Tip cot x tan x Graph y and cot x tan x y cos(x). EXAMPLE 5 cot x tan x Graph y cot x tan x. cot x tan x Simplify y. cot x tan x first Write the cotangent and tangent functions in terms of the sine and cosine functions. Simplifying Trigonometric Expressions Using Double-Angle Identities y cos x sin x sin x cos x cos x sin x sin x cos x Multiply the numerator and the denominator by sin x cos x. Recognize the double-angle and Pythagorean identities. cos x sin x y ± y cos x sin x cos x sin x y cos x sin x cos x sin x y cos(x) sin x cos x cos x sin x sin x cos x cos(x) sin x cos x a sin x cos x b Graph y cos(x). y x SECTION 5.3 SUMMARY SMH In this section, we derived the double-angle identities from the sum identities. We then used the double-angle identities to find exact values of trigonometric functions, to verify other trigonometric identities and to simplify trigonometric expressions. There is no need to memorize the second and third forms of the cosine double-angle identity since they can be derived from the first using the Pythagorean identity. sin(a) sin A cos A cos(a) cos A sin A sin A cos A tan(a) tan A tan A

32 5.3 Double-Angle Identities 87 SECTION 5.3 EXERCISES SKILLS In Exercises, use the double-angle identities to find the indicated values.. If sin x and cos x 0, find sin(x).. If sin x and cos x 0, find cos(x) If cos x 5 and sin x 0, find tan(x). 4. If cos x 5 and sin x 0, find tan(x) If tan x and p x 3p find sin(x). 6. If tan x and p x 3p find cos(x). 5, 5, 7. If sec x 5 and sin x 0, find tan(x). 8. If sec x 3 and sin x 0, find tan(x). 9. If csc x 5 and cos x 0, find sin(x). 0. If csc x 3 and cos x 0, find sin(x).. If cos x and csc x 0, find cot(x).. If sin x and cot x 0, find csc(x). 3 3 In Exercises 3, simplify each expression. Evaluate the resulting expression exactly, if possible. tan a p tan 5 8 b sin a p 6. 8 b cos a p tan 8 b 5 tan a p 8 b 7. cos (x) sin (x) 8. cos (x ) sin (x ) 9. sin sin 5 cos 5 4 sin a 3p 8 b cos a3p 8 b tan(4x).. tan (4x) cos 95 In Exercises 3 4, verify each identity. 3. csc( A) csc A sec A (sin x cos x)(cos x sin x) cos(x) cos(x) cos x cos 4 x sin 4 x cos(x) sin x cos x cos(4x) sec x sin x csc (x) sin(3x) sin x (4 cos x ) sin(4x) sin x cos x 4 sin3 x cos x 38. cot( A) (cot A tan A) (sin x cos x) sin(x) cos(x) sin x cos 4 x sin 4 x sin (x) sin (4x) cos (x) cos 4 (x) cos (x) sin (x) 4 csc(4x) sec x csc x cos(x) tan(3x) tan x(3 tan x) ( 3 tan x) cos(4x) [cos(x) sin(x)][cos(x) sin(x)] 39. sin(4x) sin(x)( 4 sin x) 40. cos(4x) sin (x) 4(sin x cos x) 4(sin x)(cos x)[cos(x)] 4. tan(4x) 4. cos(6x) [ sin x cos x cos(x) sin x] sin (x)

33 88 CHAPTER 5 Trigonometric Identities In Exercises 43 46, graph the functions. cot x tan x 43. y sin(x) 44. y tan x 45. y 46. y (tan x)(cot x)(sec x)(csc x) cos(x) sec x cot x tan x APPLICATIONS 47. Business/Economics. Annual cash flow of a stock fund (measured as a percentage of total assets) has fluctuated in cycles. The highs were roughly % of total assets and lows were roughly 8% of total assets. This cash flow can be modeled by the function C(t) 0sin t Use a double-angle identity to express C(t) in terms of the cosine function. 48. Business. Computer sales are generally subject to seasonal fluctuations. An analysis of the sales of a computer manufacturer during is approximated by the function s(t) cos t t where t represents time in quarters (t represents the end of the first quarter of 008), and s(t) represents computer sales (quarterly revenue) in millions of dollars. Use a double-angle identity to express s(t) in terms of the cosine function. 49. Hiking. Two hikers leave from the same campsite and walk in different directions. The distance d in miles between the hikers can be found using the function d 3 cos u, where u is the angle between the directions traveled by the hikers. Find a function for the distance between the hikers if u is doubled and then use a double-angle formula to write the function in terms of the cosine of a single angle u. 50. Hiking. Two hikers leave from the same campsite and walk in different directions. The distance d in miles between the hikers can be found using the function d 4 40 cos u, where u is the angle between the directions traveled by the hikers. Find a function for the distance between the hikers if u is doubled and then use a double-angle formula to write the function in terms of the sine of a single angle u. 5. Biology/Health. The rise and fall of a person s body temperature t days after contracting a certain virus can be modeled by the function T sin t, where T is body temperature in degrees Fahrenheit and 0 t 3. Write the function in terms of the cosine of a double angle and then sketch its graph. 5. Biology/Health. The rise and fall of a person s body temperature t days after contracting a certain virus can be modeled by the function T sin t cos t, where T is body temperature in degrees Fahrenheit and 0 t.5. Write the function in terms of the sine of a double angle and then sketch its graph. For Exercises 53 and 54, refer to the following: An ore crusher wheel consists of a heavy disk spinning on its axle. Its normal (crushing) force F in pounds between the wheel and the inclined track is determined by F W sin u c c C R ( cos u) A l where W is the weight of the wheel, u is the angle of the axis, C and A are moments of inertia, R is the radius of the wheel, l is the distance from the wheel to the pin where the axle is attached, and c is the speed in rpm that the wheel is spinning. The optimum crushing force occurs when the angle u is between 45 and 90. l sin u d 53. Ore-Crusher Wheel. Find F if the angle is 60, W is 500 pounds, and c C is 00 rpm, 750, and A R l 54. Ore-Crusher Wheel. Find F if the angle is 75, W is 500 pounds, and c C is 00 rpm, 750, and A R l

34 5.3 Double-Angle Identities 89 CATCH THE MISTAKE In Exercises 55 and 56, explain the mistake that is made. 55. If cos x 3, find sin( x) given sin x If sin x 3, find tan( x) given cos x 0. Write the double-angle identity for the sine function. sin(x) sin x cos x Use the quotient identity. Solve for using the sin x a sin x 3 b Use the double-angle tan(x) formula for the sine function. Pythagorean identity. sin x Cancel the common 3 cosine factors. CONCEPTUAL In Exercises 57 60, determine whether each statement is true or false. 57. sin(a) sin(a) sin(4a) 6. Let n be a positive integer. Express sin a np as a b cos anp b 58. Substitute cos x 3 and sin x 3. sin(x) 4 9 This is incorrect. What mistake was made? cos(4a) cos(a) cos(a) sin(x) a 3 b a 3 b Substitute sin x 3. This is incorrect. What mistake was made? tan(x) sin(x) cos x tan(x) 3 sin x cos x cos x tan(x) sin x single trigonometric function, and then evaluate if possible. 59. If tan x 0, then tan(x) If sin x 0, then sin(x) 0. CHALLENGE 63. Express tan(4x) in terms of functions of tan x. 64. Express tan(4x) in terms of functions of tan x. 65. Is the identity csc(x) tan x true for x p? Explain. tan x 6. Let n be a positive integer. Write the expression in terms of the cosine of a multiple angle, and then evaluate if possible. sin a np b 66. Is the identity tan(x) tan x true for x p? Explain. tan x Find all values for x where 0 x p and sin(x) sin x. 68. Find all values for x where 0 x p and cos(x) cos x. TECHNOLOGY For Exercises 69 7, refer to the following: We cannot prove that an equation is an identity using technology, but we can use technology as a first step to see whether or not the equation seems to be an identity. 69. Using a graphing calculator, plot ( x)3 ( x)5 Y ( x) and Y sin( x) for x 3! 5! ranging [, ]. Is Y a good approximation to Y? 70. Using a graphing calculator, plot ( x) ( x)4 Y and Y cos( x) for x ranging! 4! [, ]. Is a good approximation to Y? Y 7. Using a graphing calculator, determine whether tan(4 x) tan(3 x) csc( x) by plotting each side tan x sec( x) of the equation and seeing if the graphs coincide. 7. Using a graphing calculator, determine whether csc( x) sec( x) [cos( x) sin( x)] sin x sin x cos x by plotting each side of the sin x cos x(cos x sin x) equation and seeing if the graphs coincide.

35 SECTION 5.4 HALF-ANGLE IDENTITIES SKILLS OBJECTIVES Use the half-angle identities to find certain exact values of trigonometric functions. Use the half-angle identities to verify other trigonometric identities. CONCEPTUAL OBJECTIVE Understand that the half-angle identities are derived from the double-angle identities. Applying Half-Angle Identities We now use the double-angle identities from Section 5.3 to derive the half-angle identities. Like the double-angle identities, the half-angle identities will allow us to find certain exact values of trigonometric functions and to verify other trigonometric identities. We start by rewriting the second and third forms of the cosine double-angle identity to obtain identities for the square of the sine and cosine functions, sin x and cos x, otherwise known as the power reduction formulas. WORDS MATH Power reduction formula for the sine function Write the second form of the cosine double-angle identity. cos( A) sin A Isolate the sin A term on one side of the equation. sin A cos( A) Divide both sides by. sin cos( A) A Power reduction formula for the cosine function Write the third form of the cosine double-angle identity. cos( A) cos A Isolate the cos A term on one side of the equation. cos A cos(a) Divide both sides by. cos cos( A) A Power reduction formula for the tangent function Taking the quotient of these leads us to another identity. tan A sin A cos A cos( A) cos( A) These three identities for the squared functions, which are essentially alternative forms of the double-angle identities, are used in calculus as power reduction formulas (identities that allow us to reduce the power of the trigonometric function from to ): cos( A) cos( A) sin A cos( A) cos A cos( A) tan A cos( A) cos( A) 90

36 5.4 Half-Angle Identities 9 Derivation of the Half-Angle Identities We can now use these forms of the double-angle identities to derive the half-angle identities. WORDS Sine half-angle identity For the sine half-angle identity, start with the double-angle formula involving both the sine and cosine functions, cos(x) sin x, and solve for sin x. Solve for sin x. MATH sin x cos( x) sin x B cos(x) Let x A. Cosine half-angle identity For the cosine half-angle identity, start with the double-angle formula involving only the cosine functions, cos(x) cos x, and solve for cos x. Solve for cos x. Let x A. Tangent half-angle identity For the tangent half-angle identity, start with the quotient identity. sin a A b R sin a A b cos A B cos x cos x B cos(x) cos a A b R cos( x) cos a A b cos A B A tan a A sin a b b cos a A b cosa A b cosa A b Substitute half-angle identities for the sine and cosine functions. tan a A b cos A B B cos A tan a A b cos A B cos A

37 9 CHAPTER 5 Trigonometric Identities Note: can also be found by starting with the identity, tan cos( x) tan a A x, solving b cos( x) for tan x, and letting x A The tangent function also has two other similar forms for. tan a A (see Exercises 63 and 64). b HALF-ANGLE IDENTITIES FOR THE SINE, COSINE, AND TANGENT FUNCTIONS SINE COSINE TANGENT sina A b cos A B cosa A b cos A B tana A b cos A B cos A tana A b sin A cos A tana A cos A b sin A Study Tip The sign ( or ) is determined by A which quadrant contains and the sign of the particular trigonometric function in that quadrant. Applying Half-Angle Identities It is important to note that these identities hold for any real number A or any angle with either degree measure or radian measure as long as both sides of the equation are defined. The sign ( or ) is determined by the sign of the trigonometric function in the quadrant that contains A. Technology Tip Use a TI calculator to compare the values of cos 5 and 3. Be sure the calculator is B 4 in degree mode. EXAMPLE Finding Exact Values Using Half-Angle Identities Use a half-angle identity to find cos 5. Write cos 5 in terms of a half-angle. Write the half-angle identity for the cosine function. cos 5 cos a 30 b cos a A b cos A B Substitute A 30. cos a 30 b cos 30 B Classroom Example 5.4. Compute exactly sec a 5p. 8 b cos 5 R 3 Answer: 5 is in quadrant I where the cosine function is positive. cos 5 B Answer: YOUR TURN Use a half-angle identity to find sin(.5 ).

38 5.4 Half-Angle Identities 93 EXAMPLE Use a half-angle identity to find tan a p. b Finding Exact Values Using Half-Angle Identities Classroom Example 5.4.* Compute exactly cot(05 ). Answer: 3 Write tan a p in terms of a half angle. b p tan a p 6 b tan Write the half-angle identity for the tangent function.* tan a A cos A b sin A Substitute A p 6. tan p 6 cos a p 6 b sin a p 6 b tan a p 3 b Technology Tip Use a TI calculator to compare the values of tan a p and b 3. Be sure the calculator is in radian mode. tan a p b 3 p is in quadrant II where the tangent function is negative. Notice that if we approximate tan ap with a calculator, we find that tan a p b 0.679, and b *This form of the tangent half-angle identity was selected because of mathematical simplicity. If we had selected either of the other forms, we would have obtained an expression that had a square root of a square root, or a radical in the denominator. YOUR TURN Use a half-angle identity to find tan a p 8 b. Answer: or

39 94 CHAPTER 5 Trigonometric Identities Classroom Example If cot x and p x 3p, 5 find cos a x. b Answer: 6 6 EXAMPLE 3 Finding Exact Values Using Half-Angle Identities If cos x 3 3p and x p, find sin a x b, cos a x b, and tan a x 5 b. x 3p Determine in which quadrant lies. Since x p, 3p we divide the inequality by. 4 x p x lies in quadrant II; therefore, the sine function is positive and the cosine and tangent functions are negative. Write the half-angle identity for the sine function. sin a x b cos x B Substitute cos x 3 5. x Since lies in quadrant II, choose the positive value for the sine function. Write the half-angle identity for the cosine function. sin a x b R sin a x b B sin a x 5 b cos a x b cos x B Substitute cos x 3 5. x Since lies in quadrant II, choose the negative value for the cosine function. Use the quotient identity for tangent. cos a x b R cos a x b 4 B cos a x b 5 5 x tan a x sin a b b cos a x b 3 5 Answer: sin a x 5 b 5 cos a x b 5 5 Substitute sin a x 5 b 5 cos a x b 5 5. and tan a x b tan a x b tan a x b YOUR TURN If cos x 3 and p x 3p find sin a x. b, cos a x b, and tan a x 5, b

40 5.4 Half-Angle Identities 95 EXAMPLE 4 Using Half-Angle Identities to Verify Other Trigonometric Identities Verify the identity cos a x tan x sin x b. tan x Technology Tip Graph y cos a x and b Start with the left side. Write the cosine half-angle identity. Square both sides of the equation. Multiply the numerator and the denominator on the right side by tan x. Note that cos x tan x sin x. cos a x b cos x B cos a x cos x b cos a x cos x b a b a tan x tan x b cos a x tan x cos x tan x b tan x cos a x tan x sin x b tan x y tan x sin x. tan x An alternative solution is to start with the right-hand side. Solution (alternative): Start with the right-hand side. Write this expression as the sum of two expressions. Write tan x sin x. cos x tan x sin x tan x tan x sin x tan x tan x sin x sin x cos x ( cos x) cos a x b sin x tan x Classroom Example Verify that cot a A b csc aa b tan aa b csc A. cos a A b Answer: Use basic identities to rewrite all expressions in terms of sin aa and cos a A. Then use b b the double-angle identity. EXAMPLE 5 Using Half-Angle Identities to Verify Other Trigonometric Identities Verify the identity tan x csc( x) cot( x). Notice that x is of x. Write the third half-angle formula for the tangent function. Write the right side as a difference of two expressions having the same denominator. Substitute the reciprocal and quotient identities, respectively, on the right. tan a A cos A b sin A tan a A b cos A sin A sin A tan a A b csc A cot A Let A x. tan x csc( x) cot( x) Classroom Example Graph y cos(5x). B Answer: First, reduce this to y cosa5x b. 3 4

41 96 CHAPTER 5 Trigonometric Identities Notice that in Example 5, we started with the third half-angle identity for the tangent function. In Example 6, we will start with the second half-angle identity for the tangent function. In general, you select the form that appears to lead to the desired expression. Technology Tip Graph y sin(px) and cos(px) y tan(px). Answer: y ( y = tan x x ) EXAMPLE 6 Using Half-Angle Identities to Simplify Trigonometric Expressions Graph y sin(px) by first simplifying the trigonometric expression to a more cos(px) recognizable form. Simplify the trigonometric expression using a half-angle identity for the tangent function. Write the second half-angle identity for the tangent function. Let A px. Graph y tan(px). cos(p x) YOUR TURN Graph y. sin(p x) tan a A b sin A cos A tan(p x) sin(px) cos(px) y = tan(x) y = sin(x) + cos(x) y x SECTION 5.4 SUMMARY SMH In this section, we used the double-angle identities to derive the half-angle identities. We then used the half-angle identities to find certain exact values of trigonometric functions, verify other trigonometric identities, and simplify trigonometric expressions. sin a A b cos A B cos a A b cos A B The sign ( or ) is chosen by first determining which A quadrant contains and then determining the sign of the indicated trigonometric function in that quadrant. Recall that there are three forms of the tangent half-angle identity. There is no need to memorize the other forms of the tangent half-angle identity, since they can be derived by first using the Pythagorean identity and algebraic manipulation. tan a A b cos A B cos A

42 5.4 Half-Angle Identities 97 SECTION 5.4 EXERCISES SKILLS In Exercises 6, use the half-angle identities to find the exact values of the trigonometric expressions.. sin 5. cos(.5 ) 3. cos ap 4. b 5. cos sin cos a3p 8. b 9. cos a3p 0. sin a 7p. tan(67.5 ). 8 b b 3. sec a9p 4. csc a9p 5. cot a3p 6. 8 b 8 b 8 b sin a p 8 b sin a 9p 8 b tan(0.5 ) cot a 7p 8 b In Exercises 7 6, use the half-angle identities to find the desired function values. 5 3p 7. If cos x and x p, find sin a x 8. If cos x 5 and p x 3p find cos a x 3 b. 3, b. 9. If tan x and p x 3p find sin a x 0. If tan x and p x 3p find cos a x 5, b. 5, b. 3p. If sec x 5 and 0 x p find tan a x. If sec x 3 and x p, find tan a x, b. b. p 3p 3. If csc x 3 and x p, find sin a x 4. If csc x 3 and x p, find cos a x b. b. 5. If cos x and p x 3p 3p find cot a x 6. If cos x and x p, find csc a x 4, b. 4 b. In Exercises 7 30, simplify each expression using half-angle identities. Do not evaluate. cos a 5p cos a p 6 b 4 b sin R R cos 50 In Exercises 3 44, verify the identities. cos 50 sin sin a x b cos a x b sin a x sin(90 x) b sin a x b cos x sin x cos a x b sin x tan a x cos x b cos x tan aa 40. b cot aa b csc A cos a x b sin a x b cos x sin a x b cos a x b sin x cos a x b sin x tan a x b (csc x cot x) cot a A b tan aa b cot A sec a A ( cos A) csc a A ( cos A) b b sin A sin A

43 98 CHAPTER 5 Trigonometric Identities 43. csc aa b ƒ csc A ƒ cos A 44. sec a A b ƒ csc A ƒ cos A In Exercises 45 48, graph the functions. tan a x y c sin a x b cos a x b y 4 y 6 sin a x cos a x y b b bd tan a x b APPLICATIONS 49. Area of an Isosceles Triangle. Consider the triangle below, where the vertex angle measures u, the equal sides measure a, the height is h, and half the base is b. (In an isosceles triangle, the perpendicular dropped from the vertex angle divides the triangle into two congruent triangles.) The two triangles formed are right triangles. In the right triangles, sin au and cos a u b h b b a a. Multiply each side of each equation by a to get b a sin a u. The area of the entire b, h a cos au b isosceles triangle is A (b)h bh. Substitute the values for b and h into the area formula. Show that the area is a equivalent to. sin u 50. Area of an Isosceles Triangle. Use the results from Exercise 49 to find the area of an isosceles triangle whose equal sides measure 7 inches and whose base angles each measure Biking. A bicycle ramp is made so that it can easily be raised and lowered for different levels of competition. For the advance division, the angle formed by the ramp and the 5 ground is u such that tan u, the steepness of the ramp, is 3. For the novice division, the angle u is cut in half to lower u the ramp. What is the steepness of the ramp for angle? 5. Biking. A bicycle ramp is made so that it can easily be raised and lowered for different levels of competition. For the advance division, the angle formed by the ramp and the ground is u such that sin u. For the novice 3 division, the angle u is cut in half to lower the ramp. What is tan a u the steepness of the ramp? b, a b h a 53. Farming. An auger used to deliver grain to a storage bin can be raised and lowered, thus allowing for different size bins. Let a be the angle formed by the auger and the ground for bin A such that sin a The angle formed by the auger and the ground for bin B is half of a. If the height h, in feet, of a bin can be found using the formula h 75 sin u, where u is the angle formed by the ground and the auger, find the height of bin B. 54. Farming. An auger used to deliver grain to a storage bin can be raised and lowered, thus allowing for different size bins. Let a be the angle formed by the auger and the ground for bin A such that tan a 4 7. The angle formed by the auger and the ground for bin B is half of a. If the height h, in feet, of a bin can be found using the formula h 75 sin u, where u is the angle formed by the ground and the auger, find the height of bin B. For Exercises 55 and 56, refer to the following: Monthly profits can be expressed as a function of sales, that is, p(s). A financial analysis of a company has determined that the sales s in thousands of dollars are also related to monthly profits p in thousands of dollars by the relationship: tan u p for 0 s 50, 0 p 40 s Based on sales and profits, it can be determined that the domain for angle u is 0 u 38. Profits (p) p Sales (s) s 55. Business. If monthly profits are $3000 and monthly sales are $4000, find tan a u. b 56. Business. If monthly profits are p and monthly sales are s (where p s), find tan a u. b p(s)

44 5.4 Half-Angle Identities 99 CATCH THE MISTAKE In Exercises 57 and 58, explain the mistake that is made. 57. If cos x, find sin a x given p x 3p 3 b. 58. If cos x find tan a x. 3, b Write the half-angle identity for the sine function. Substitute cos x 3. sin a x b cos x B sin a x b 3 R sin a x b B 3 Use the quotient identity. Use the half-angle identity for sine. tan a x sin a x b b tan a x b cos x cos x cos x tan a x b a cos x cos x cos x b The sine function is negative. sin a x b B 3 tan a x b a cos x cos x b This is incorrect. What mistake was made? Substitute cos x 3. tan a x b tan a x b This is incorrect. What mistake was made? CONCEPTUAL In Exercises 59 6, determine whether each statement is true or false sin a A b sin aa b sin A cos a A b cos aa b cos A 6. If tan x 0, then 6. If sin x 0, then sin a x b 0. tan a x b Given tan a A verify that b cos A B cos A, tan a A Substitute A p into the identity b sin A cos A. and explain your results. 64. Given tan a A verify that b cos A B cos A, tan a A cos A b Substitute A p into the identity sin A. and explain your results.

45 300 CHAPTER 5 Trigonometric Identities CHALLENGE 65. Find the exact value of sin 5 in two ways, using sum and difference identities and half-angle identities; then show that they are equal. 66. Find the exact value of tan 5 in two ways, using sum and difference identities and half-angle identities; then show that they are equal. 67. Express tan a x in terms of the cosine of a single angle. 4 b 68. Express sin a x in terms of the cosine of a single angle. 4 b TECHNOLOGY For Exercises 69 7, refer to the following: One cannot prove that an equation is an identity using technology, but one can use it as a first step to see whether the equation seems to be an identity. 3 x 69. Using a graphing calculator, plot Y a x a b b a x 5 b 3! 5! and Y sin a x for x range [, ]. Is Y a good b approximation to Y? a x b a x 4 b 70. Using a graphing calculator, plot Y! 4! and Y cos a x for x range [, ]. Is a good b Y 7. Using a graphing calculator, determine whether csc a x is an identity by plotting b sec a x b 4 csc x each side of the equation and seeing if the graphs coincide. 7. Using a graphing calculator, determine whether tan a x is an identity by b cot a x b cot x sec x plotting each side of the equation and seeing if the graphs coincide. approximation to Y? SECTION 5.5 PRODUCT-TO-SUM AND SUM-TO-PRODUCT IDENTITIES SKILLS OBJECTIVES Express products of trigonometric functions as sums of trigonometric functions. Express sums of trigonometric functions as products of trigonometric functions. CONCEPTUAL OBJECTIVES Understand that the sum and difference identities are used to derive the product-to-sum identities. Understand that the product-to-sum identities are used to derive the sum-to-product identities. Often in calculus it will be helpful to write products of trigonometric functions as sums of other trigonometric functions and vice versa. In this section, we discuss the product-to-sum identities, which convert products of trigonometric functions to sums of trigonometric functions, and sum-to-product identities, which convert sums of trigonometric functions to products of trigonometric functions.

46 5.5 Product-to-Sum and Sum-to-Product Identities 30 Product-to-Sum Identities The product-to-sum identities are derived from the sum and difference identities. WORDS Write the identity for the cosine of a sum. Write the identity for the cosine of a difference. Add the two identities. Divide both sides by. Subtract the sum identity from the difference identity. Divide both sides by. Write the identity for the sine of a sum. Write the identity for the sine of a difference. Add the two identities. Divide both sides by. MATH cos A cos B sin A sin B cos(a B) cos A cos B sin A sin B cos(a B) cos A cos B cos(a B) cos(a B) cos A cos B [cos(a B) cos(a B)] cos A cos B sin A sin B cos(a B) cos A cos B sin A sin B cos(a B) sin A sin B cos(a B) cos(a B) sin A sin B [cos(a B) cos(a B)] sin A cos B cos A sin B sin(a B) sin A cos B cos A sin B sin(a B) sin Acos B sin(a B) sin(a B) sin A cos B [sin(a B) sin(a B)] PRODUCT-TO-SUM IDENTITIES. cos A cos B [cos(a B) cos(a B)]. sin A sin B [cos(a B) cos(a B)] 3. sin A cos B [sin(a B) sin(a B)] EXAMPLE Illustrating a Product-to-Sum Identity for Specific Values Show that product-to-sum identity (3) is true when A 30 and B 90. Write product-to-sum identity (3). Let A 30 and B 90. Evaluate the trigonometric functions. 0 0 sin A cos B [sin(a B) sin(a B)] sin 30 cos 90 [sin(30 90 ) sin(30 90 )] sin 30 cos 90 [sin 0 sin(60 )] 0 c 3 3 d Classroom Example 5.5. Show that the product-to-sum identity () is true when A 45 and B 5. Answer: cos A cos B [cos(a B) cos(a B)] cos 45 cos 5 [cos(45 5 ) cos(45 5 )] [cos 70 cos(80 )] a b [0 ()]

47 30 CHAPTER 5 Trigonometric Identities Technology Tip Graph y cos(4x) cos(3x) and y [cos(7x) cos x ]. EXAMPLE Converting a Product to a Sum Convert the product cos(4x) cos(3x) to a sum. Write the product-to-sum identity (). cos A cos B [cos(a B) cos(a B)] Let A 4x and B 3x. cos(4 x) cos(3x) [cos(4 x 3x) cos(4 x 3x)] cos(4 x) cos(3x) [cos(7x) cos x] Answer: [cos(7x) cos(3x)] YOUR TURN Convert the product cos(x) cos(5x) to a sum. Classroom Example 5.5. Convert sin(px)cos a p to 4 xb a sum. Answer: c sin a5p 4 xb sin a3p 4 xbd EXAMPLE 3 Converting Products to Sums Express sin(x) sin(3x) in terms of cosines. Write the product-to-sum identity (). Let A x and B 3x. sin A sin B [cos(a B) cos(a B)] sin(x) sin(3x) [cos(x 3x) cos(x 3x)] The cosine function is an even function. sin(x) sin(3x) sin(x) sin(3x) [cos(x) cos(5x)] [cos x cos(5x)] Answer: [cos x cos(3x)] Classroom Example 5.5.3* Express sin(npx) sin(mpx) in terms of cosines. Here, n and m are integers. Answer: {cos[(n m)px] cos[(n m)px]} YOUR TURN Express sin x sin(x) in terms of cosines. Sum-to-Product Identities The sum-to-product identities can be obtained from the product-to-sum identities. WORDS Write the identity for the product of the sine and cosine functions. Let x y A and x y B. Solve for x and y in term of A and B. Substitute these values into the identity. Multiply by. MATH [sin(x y) sin(x y)] sin x cos y x y A x y B x A B x A B (sin A sin B) sin aa B x y A x y B y A B sin A sin B sin a A B y A B b cos a A B b b cos a A B b

48 5.5 Product-to-Sum and Sum-to-Product Identities 303 The other four sum-to-product identities can be found similarly. All are summarized in the following box: SUM-TO-PRODUCT sin A sin B sin a A B sin A sin B sin a A B cos A cos B cos a A B 7. cos A cos B sin a A B IDENTITIES b cos a A B b b cos a A B b b cos a A B b b sin a A B b EXAMPLE 4 Illustrating a Sum-to-Product Identity for Specific Values Show the sum-to-product identity (7) is true when A 30 and B 90. Write the sum-to-product identity (7). Let A 30 and B 90. The sine function is an odd function. Evaluate the trigonometric functions. cos A cos B sin a A B cos 30 cos 90 sin a b sin a b cos 30 cos 90 sin 60 sin(30 ) cos 30 cos 90 sin 60 sin b sin a A B b Classroom Example Convert the sum 6[cos(5x) cos(35x)] to a product. Answer: cos(5x) cos(5x) EXAMPLE 5 Converting a Sum to a Product Convert 9[sin(x) sin(0x)] to a product. Technology Tip Graph y 9[sin(x) sin(0x)] and y 8 sin(4x) cos(6x). The expression inside the brackets is in the form of identity (5). Let A x and B 0x. The sine function is an odd function. sin A sin B sin a A B x 0x x 0x sin(x) sin(0x) sin a b cos a b sin( x) sin(0 x) sin(4 x) cos(6 x) sin( x) sin(0 x) sin(4 x) cos(6 x) b cos a A B b Multiply both sides by 9. 9[sin( x) sin(0 x)] 8 sin(4 x) cos(6 x)

49 304 CHAPTER 5 Trigonometric Identities Classroom Example Simplify sin( x) cos(x ). Answer: [sin(x 3) sin( 3x)] Classroom Example 5.5.6* Simplify cos(3x 4y) cos(3x 4y). sin(3x 4y) sin(3x 4y) Answer: tan(4y) EXAMPLE 6 Simplifying a Trigonometric Expression Using Sum-to-Product Identities Simplify the expression sin a x y b cos a x y b sin a x y b cos a x y b. Use identities (4) and (5). sin a x y b cos a x y b sin a x y b cos a x y b sin x sin y sin x sin y sin x (sin x sin y) (sin x sin y) Applications In music, a note is a fixed pitch (frequency) that is given a name. If two notes are sounded simultaneously, then they combine to produce another note often called a beat. The beat frequency is the difference of the two frequencies. The more rapid the beat, the further apart the two frequencies of the notes are. When musicians tune their instruments, they use a tuning fork to sound a note and then tune the instrument until the beat is eliminated; hence, the fork and instrument are in tune with each other. Mathematically, a note or tone is represented as A cos(pft), where A is the amplitude (loudness), f is the frequency in hertz, and t is time in seconds. The following figure summarizes common notes and frequencies: C 6 Hz D 94 Hz E 330 Hz F 349 Hz G 39 Hz A 440 Hz B 494 Hz Music EXAMPLE 7 Express the musical tone when a C and G are simultaneously struck (assume with the same loudness). Find the beat frequency f f. Assume uniform loudness, A. Write the mathematical description of a C note. Write the mathematical description of a G note. Add the two notes. Use a sum-to-product identity: cos(54t) cos(784t). Cosine is an even function: cos(x) cosx. Identify average frequency and beat of the tone. cos(pf t), f 6 Hz cos(pf t), f 39 Hz cos(54pt) cos(784pt) 54pt 784pt 54pt 784pt cos a b cos a b cos(654pt) cos(30pt) cos(654pt) cos(30pt) cos(37t) cos(30t) average beats per frequency second

50 5.5 Product-to-Sum and Sum-to-Product Identities 305 The beat frequency can also be found by subtracting f from f. Therefore, the tone of average frequency, 37 hertz, has a beat of 30 hertz (beats/per second). f f Hz y = cos(54t) y y = cos(784t) t y = cos(654t) cos(30t) SECTION 5.5 SUMMARY In this section, we used the sum and difference identities (Section 5.) to derive the product-to-sum identities. The product-to-sum identities allowed us to express products of trigonometric functions as sums of trigonometric functions. cos A cos B [cos(a B) cos(a B)] sin A sin B [cos(a B) cos(a B)] sin A cos B [sin(a B) sin(a B)] We then used the product-to-sum identities to derive the sum-to-product identities. The sum-to-product identities allow us to express sums as products. sin A sin B sin a A B sin A sin B sin a A B cos A cos B cos a A B b cos a A B b b cos a A B b b cos a A B b cos A cos B sin a A B b sin a A B b SECTION 5.5 EXERCISES SKILLS In Exercises, write each product as a sum or difference of sines and/or cosines.. sin( x) cos x. cos(0 x) sin(5 x) 3. 5 sin(4 x) sin(6 x) cos(x) cos(x) 6. 8 cos(3 x) cos(5 x) 7. sin a3xb sin a5x 8. b 9. cos axb cos a4x 0. sin a p. sin(97.5 ) sin(.5 ). 3 3 b 4 xb cos ap xb In Exercises 3 4, write each expression as a product of sines and/or cosines. 3. cos(5 x) cos(3 x) 4. cos( x) cos(4 x) 5. sin(3 x) sin x sin(8x) sin(6x) 8. cos(x) cos(6x) 9. sin a x b sin a5x 0. b 3 sin( x) sin(4 x) sin a px b sin a5px b cos(85.5 ) cos(4.5 ) sin(0 x) sin(5 x) cos a x b cos a5x b. cos a. sin a 3. sin(0.4 x) sin(0.6 x) 4. cos(0.3 x) cos(0.5 x) 3 xb cos a7 3 xb 3 xb sin a7 3 xb

51 306 CHAPTER 5 Trigonometric Identities In Exercises 5 8, simplify the trigonometric expressions. cos(3 x) cos x sin(4 x) sin( x) cos x cos(3 x) sin(3 x) sin x cos(4 x) cos( x) sin(3 x) sin x sin(4 x) sin( x) cos(4 x) cos( x) In Exercises 9 38, verify the identities. sin A sin B B 9. tan aa b 30. cos A cos B cos A cos B B 3. tan aa b 3. sin A sin B sin A sin B B 33. tan aa b cot a A B b 34. sin A sin B 35. sin A sin B [ cos(a B)][ cos(a B)] cos A cos B cos A cos B csin aa b cos ab b cos aa b sin ab bd csin aa b cos ab b cos aa b sin ab bd 38. sin A sin a A B b cos a A B b sin a A B b cos a A B b sin A sin B cos A cos B cos A cos B sin A sin B cos A cos B cos A cos B B tan aa b B tan aa b tan aa B b tan a A B b sin A cos B cos A sin B sin(a B) APPLICATIONS 39. Business. An analysis of the monthly costs and monthly revenues of a toy store indicates that monthly costs fluctuate (increase and decrease) according to the function C(t) sina p t pb 6 and monthly revenues fluctuate (increase and decrease) according to the function R(t) sin a p 6 t 5p 3 b Find the function that describes how the monthly profits fluctuate: P(t) R(t) C(t). Using identities in this section, express P(t) in terms of a cosine function. 40. Business. An analysis of the monthly costs and monthly revenues of an electronics manufacturer indicates that monthly costs fluctuate (increase and decrease) according to the function C(t) cos a p 3 t p 3 b and monthly revenues fluctuate (increase and decrease) according to the function R(t) cos a p 3 tb Find the function that describes how the monthly profits fluctuate: P(t) R(t) C(t). Using identities in this section, express P(t) in terms of a sine function. 4. Music. Write a mathematical description of a tone that results from simultaneously playing a G and a B. What is the beat frequency? What is the average frequency? 4. Music. Write a mathematical description of a tone that results from simultaneously playing an F and an A. What is the beat frequency? What is the average frequency? 43. Optics. Two optical signals with uniform (A ) intensities and wavelengths of.55 micrometer and 0.63 micrometer are beat together. What is the resulting sum if their individual signals are given by sin a ptc and sin a ptc.55m b, where c meters per second? 0.63m b (Note: m 0 6 m.) 44. Optics. The two optical signals in Exercise 43 are beat together. What are the average frequency and the beat frequency? For Exercises 45 and 46, refer to the following: Touch-tone keypads have the following simultaneous low and high frequencies: FREQUENCY 09 Hz 336 Hz 477 Hz 697 HZ HZ HZ Hz * 0 # The signal made when a key is pressed is sin(pf t) sin(pf t), where f is the low frequency and f is the high frequency. 45. Touch-Tone Dialing. What is the mathematical function that models the sound of dialing 4?

52 5.5 Product-to-Sum and Sum-to-Product Identities Touch-Tone Dialing. What is the mathematical function that models the sound of dialing 3? 47. Area of a Triangle. A formula for finding the area of a triangle when given the measures of the angles and one side is area a sin B sin C, where a is the side opposite sin A angle A. If the measures of angles B and C are 5.5 and 7.5, respectively, and if a 0 feet, use the appropriate product-to-sum identity to change the formula so that you can solve for the area of the triangle exactly. B A 48. Area of a Triangle. If the measures of angles B and C in Exercise 45 are 75 and 45, respectively, and if a inches, use the appropriate product-to-sum identity to change the formula so that you can solve for the area of the triangle exactly. B a C 49. Calculus. In calculus, there is an operation called integration that serves a number of purposes. When performing integration on trigonometric functions, it is much easier if the expression contains a single trigonometric function or the sum of trigonometric functions instead of the product of trigonometric functions. Using identities, change the following expression so that it does not contain the product of trigonometric functions. sin A sin B sin A cos B cos A cos B 50. Calculus. In calculus, there is an operation called integration that serves a number of purposes. When performing integration on trigonometric functions, it is much easier if the expression contains a single trigonometric function or the sum of trigonometric functions instead of the product of trigonometric functions. Using identities, change the following expression so that it does not contain the product of trigonometric functions. cos a A B bcsin a A B b cos a A B bd a A C CATCH THE MISTAKE In Exercises 5 and 5, explain the mistake that is made. 5. Simplify the expression (cos A cos B) (sin A sin B). Expand by squaring. Regroup terms. cos A cos A cos B cos B sin A sin A sin B sin B cos A sin A cos A cos B sin A sin B cos B sin B Simplify using the Pythagorean identity. cos A sin A cos A cos B sin A sin B cos B sin B Factor the common. ( cos A cos B sin A sin B) [ cos(ab) sin(ab)] This is incorrect. What mistake was made? 5. Simplify the expression (sin A sin B)(cos A cos B). Multiply the expressions using the distribution property. sin A cos A sin A cos B sin B cos A sin B cos B Cancel the second and third terms. Use the product-to-sum identity. sin A cos A [sin(a A) sin(a A) ] sin( A) sin( B) This is incorrect. What mistake was made? sin A cos A sin B cos B sin B cos B [sin(b B) sin(b B) ]

53 308 CHAPTER 5 Trigonometric Identities CONCEPTUAL In Exercises 53 56, determine whether each statement is true or false cos A cos B cos(ab) sin A sin B sin(ab) 57. Write sin A sin B sin C as a sum or difference of sines and cosines. 58. Write cos A cos B cos C as a sum or difference of sines and 55. The product of two cosine functions is a sum of two other cosines. cosine functions. 56. The product of two sine functions is a difference of two cosine functions. CHALLENGE 59. Find all values of A and B such that cos A cos B cos(a B). 60. Find all values of A and B such that sin A cos B sin(a B). 6. Let A B and then verify that the identity for sin A cos B still holds true. 6. Let A B and then verify that the identity for sin A sin B still holds true. 63. Find an expression in terms of cosine for if A and B are complementary angles. sin A sin B 64. Find an expression in terms of cosine for cos A cos B if A and B are complementary angles. TECHNOLOGY 65. Suggest an identity 4 sin x cos x cos( x) by graphing Y 4 sin x cos x cos( x) and determining the function based on the graph. 66. Suggest an identity tan x tan( x) by graphing Y tan x tan(x) and determining the function based on the graph.

54 CHAPTER 5 INQUIRY-BASED LEARNING PROJECT When it comes to identities, don t always let your intuition be your guide.. Suppose a fellow student claims that the equation sin(a b) sin a sin b is an identity. He s wondering if you can help because he s not sure how to verify his claim, but says, It just seems intuitively so. Can you help this student? a. First, let s understand the student s claim. What does it mean to say that sin(a b) sin a sin b is an identity? b. If you decided to try and verify the student s claim, you d start with one side of his equation and try to manipulate that side until it looks like the other side. But, that may turn out to be a lot of unnecessary work if, in fact, the student s claim is false. So, instead, try something else. Consider a right triangle with angles a and b. Calculate the values in the chart below, using various values of a and b. b a a b a b sin(a b) sina sin b sin a sin b c. What does your data tell you about the student s claim? Explain. d. In this example, you discovered that function notation is not distributive. Now, show the student how to write the sum identity for the sine function; that is, sin(a b) = (This identity is derived in this chapter.). Word has gotten out that you are really good at helping others understand trigonometric identities. Another of your fellow students asks whether sin(a) sina is an identity. a. How many values of a would you need to check to determine whether the student s equation is an identity? Explain. b. Show how to convince the student that his equation is not an identity. c. Try to discover the double-angle identity sin(a) = : For the right triangle below, fill out the chart (exact values) and look for a pattern. a sin(a) sin a cos a a

55 MODELING OUR WORLD Sometimes data can have an oscillatory trend on the micro scale and a linear trend on the macro scale and vice versa (linear trend on the micro scale and an oscillatory trend on the macro scale). Some argue that global warming is a natural cyclic effect, and although in our lifetime it appears to be increasing, it is in fact oscillatory over longer time periods and will self-correct. Others claim the opposite is true that temperatures appear oscillatory on short time scales but overall are increasing. What do you think? The following table summarizes average yearly temperature in degrees Fahrenheit ( F) and carbon dioxide emissions in parts per million (ppm) for Mauna Loa, Hawaii. Year Temperature ( F) CO Emissions (ppm) Plot the data for temperature in the Cartesian plane with the year along the horizontal axis and temperature along the vertical axis. Let t 0 correspond to 960. Look back at the Technology Tip in the margin next to Example in Section 5.5. The left side (negative x-values) of the graph is an example of oscillatory behavior on the micro scale and increasing (linear) trend on the macro scale. Look at the result in Example : The product of two cosine functions is equal to a constant times a sum of cosine functions.. Find a function of the form f(t) = k Acos(Bt) cos(ct) that models the temperature in Mauna Loa. Assume A, k, B, and C are constants to be determined. 3. Find a function of the form f(t) = k A{cos[(B + C)t] + cos[(b C)t]} that models the temperature in Mauna Loa. 4. Compare the models developed in Questions and 3. Do they agree? Should they agree according to Example in Section 5.5? 5. Compare the models developed in Questions and 3 to the model developed in the Modeling Our World in Chapter 4 of the form f(t) = Dt + A sin(bt). 6. Which of your models do you think best fits the data? 7. Do your models support the claim of global warming? Explain. (In other words, does a strictly sinusoidal model support the claim of global warming? Does a combination of linear and sinusoidal models support the claim of global warming?) 30