3330_050.qxd /5/05 9:5 AM Page 374 374 Chapter 5 Analytic Trigonometry 5. Using Fundamental Identities What you should learn Recognize and write the fundamental trigonometric identities. Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions. Why you should learn it Fundamental trigonometric identities can be used to simplify trigonometric expressions. For instance, in Exercise 99 on page 38, you can use trigonometric identities to simplify an expression for the coefficient of friction. Introduction In Chapter 4, you studied the basic definitions, properties, graphs, and applications of the individual trigonometric functions. In this chapter, you will learn how to use the fundamental identities to do the following.. Evaluate trigonometric functions.. Simplify trigonometric expressions. 3. Develop additional trigonometric identities. 4. Solve trigonometric equations. Fundamental Trigonometric Identities Reciprocal Identities sin u csc u csc u sin u Quotient Identities tan u sin u cos u Pythagorean Identities sin u cos u Cofunction Identities sin u cos u tan u cot u sec u csc u Even/Odd Identities sinu sin u cscu csc u cos u sec u sec u cos u cot u cos u sin u tan u sec u cos u sin u cot u tan u csc u sec u cosu cos u secu sec u tan u cot u cot u tan u cot u csc u tanu tan u cotu cot u The HM mathspace CD-ROM and Eduspace for this text contain additional resources related to the concepts discussed in this chapter. Pythagorean identities are sometimes used in radical form such as sin u ± cos u or tan u ±sec u where the sign depends on the choice of u.
3330_050.qxd /5/05 9:5 AM Page 375 Section 5. Using Fundamental Identities 375 You should learn the fundamental trigonometric identities well, because they are used frequently in trigonometry and they will also appear later in calculus. Note that u can be an angle, a real number, or a variable. You can use a graphing utility to check the result of Example. To do this, graph and in the same viewing window, as shown below. Because Example shows the equivalence algebraically and the two graphs appear to coincide, you can conclude that the expressions are equivalent. π Technology y sin x cos x sin x y sin 3 x Remind students that they must use an algebraic approach to prove that two expressions are equivalent. A graphical approach can only confirm that the simplification found using algebraic techniques is correct. π Using the Fundamental Identities One common use of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions. Example Using Identities to Evaluate a Function Use the values sec u 3 and tan u > 0 to find the values of all six trigonometric functions. Using a reciprocal identity, you have cos u sec u 3 3. Using a, you have sin u cos u Substitute Simplify. for cos u. Because sec u < 0 and tan u > 0, it follows that u lies in Quadrant III. Moreover, because sin u is negative when u is in Quadrant III, you can choose the negative root and obtain sin u 53. Now, knowing the values of the sine and cosine, you can find the values of all six trigonometric functions. sin u 5 3 cos u 3 tan u sin u 53 5 cos u 3 Example Now try Exercise. Simplify sin x cos x sin x. 3 4 5 9 9. Simplifying a Trigonometric Expression First factor out a common monomial factor and then use a fundamental identity. sin x cos x sin x sin xcos x sin x cos x sin xsin x sin 3 x Now try Exercise 45. csc u sin u 3 5 35 5 sec u cos u 3 cot u tan u Factor out common monomial factor. Factor out. Multiply. 3 5 5 5
3330_050.qxd /5/05 9:5 AM Page 376 376 Chapter 5 Analytic Trigonometry When factoring trigonometric expressions, it is helpful to find a special polynomial factoring form that fits the expression, as shown in Example 3. Example 3 Factoring Trigonometric Expressions Factor each expression. a. sec b. 4 tan tan 3 a. Here you have the difference of two squares, which factors as sec sec sec ). b. This expression has the polynomial form ax bx c, and it factors as 4 tan tan 3 4 tan 3tan. Now try Exercise 47. On occasion, factoring or simplifying can best be done by first rewriting the expression in terms of just one trigonometric function or in terms of sine and cosine only. These strategies are illustrated in Examples 4 and 5, respectively. Example 4 Factoring a Trigonometric Expression Factor csc x cot x 3. Use the identity csc x cot x to rewrite the expression in terms of the cotangent. csc x cot x 3 cot x cot x 3 cot x cot x Combine like terms. cot x cot x Factor. Now try Exercise 5. Example 5 Simplifying a Trigonometric Expression Remember that when adding rational expressions, you must first find the least common denominator (LCD). In Example 5, the LCD is sin t. Simplify sin t cot t cos t. Begin by rewriting cot t in terms of sine and cosine. sin t cot t cos t sin t cos t sin t cos t sin t cos t sin t sin t csc t Now try Exercise 57. Quotient identity Add fractions. Reciprocal identity
3330_050.qxd /5/05 9:5 AM Page 377 Section 5. Using Fundamental Identities 377 Example 6 Adding Trigonometric Expressions Perform the addition and simplify. sin cos cos sin sin cos sin sin (cos cos cos sin cos sin sin cos cos cos sin cos cos sin sin csc Now try Exercise 6. Multiply. : sin cos Divide out common factor. Reciprocal identity The last two examples in this section involve techniques for rewriting expressions in forms that are used in calculus. Example 7 Rewriting a Trigonometric Expression Rewrite sin x so that it is not in fractional form. From the cos x sin x sin x sin x, you can see that multiplying both the numerator and the denominator by sin x will produce a monomial denominator. sin x sin x sin x sin x sin x sin x sin x cos x sin x cos x cos x sin x cos x cos x cos x sec x tan x sec x Now try Exercise 65. Multiply numerator and denominator by sin x. Multiply. Write as separate fractions. Product of fractions Reciprocal and quotient identities
3330_050.qxd /5/05 9:5 AM Page 378 378 Chapter 5 Analytic Trigonometry Example 8 Trigonometric Substitution Activities. Simplify, using the fundamental trigonometric identities. cot csc Answer: cos. Use the trigonometric substitution x 4 sec to rewrite the expression x 6 as a trigonometric function of, where 0 < <. Answer: 4 tan Use the substitution x tan, 0 < 4 x as a trigonometric function of. <, to write Begin by letting x tan. Then, you can obtain 4 x 4 tan 4 4 tan 4 tan 4 sec sec. Now try Exercise 77. Substitute tan Rule of exponents Factor. sec > 0 for 0 < for x. < 4 + x θ = arctan x x Angle whose tangent is. FIGURE 5. x Figure 5. shows the right triangle illustration of the trigonometric substitution x tan in Example 8. You can use this triangle to check the solution of Example 8. For 0 < <, you have opp x, adj, and hyp 4 x. With these expressions, you can write the following. sec hyp adj 4 x sec sec 4 x So, the solution checks. Example 9 Rewriting a Logarithmic Expression Rewrite ln csc ln tan ln csc ln tan ln csc tan Now try Exercise 9. as a single logarithm and simplify the result. ln sin sin cos ln cos ln sec Product Property of Logarithms Reciprocal and quotient identities Simplify. Reciprocal identity
Section 5. Using Fundamental Identities 379 5. Exercises The HM mathspace CD-ROM and Eduspace for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blank to complete the trigonometric identity.. sin u. cos u sec u 3. 4. tan u sin u 5. csc u 6. tan u 7. 8. sec u u 9. cosu 0. tanu PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.eduspace.com. In Exercises 4, use the given values to evaluate (if possible) all six trigonometric functions... 3. 4. 5. 6. cot 3, 7. sec 3, 8. sin x 3, tan x 3 3, sec, csc 5 tan 3 3, 4 tan x 5 sec x 3, cos x 3 5, 9. sinx 3, cos x cos x 3 sin sin 0 0 csc 35 5 cos x 4 5 tan x 4 0. sec x 4, sin x > 0. tan, sin < 0. csc 5, 3. sin, cos < 0 cot 0 4. tan is undefined, sin > 0 In Exercises 5 0, match the trigonometric expression with one of the following. (a) sec x (b) (c) cot x (d) (e) tan x (f) sin x 5. sec x cos x 6. tan x csc x 7. cot x csc x 8. cos xcsc x sinx 9. 0. cosx sin x cos x In Exercises 6, match the trigonometric expression with one of the following. (a) csc x (b) tan x (c) sin x (d) sin x tan x (e) sec x (f) sec x tan x. sin x sec x. cos xsec x 3. sec 4 x tan 4 x 4. cot x sec x sec x cos x 5. 6. sin x cos x In Exercises 7 44, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. 7. cot sec 8. cos tan 9. sin csc sin 30. sec x sin x 3. cot x csc 3. csc x sec 33. sin x 34. csc x tan x tan 35. sec sin 36. tan sec 37. 38. cot x sec x x cos x 39. cos y sin y 40. cos t tan t 4. sin tan cos 4. csc tan sec 43. cot u sin u tan u cos u 44. sin sec cos csc
80 Chapter 5 Analytic Trigonometry n Exercises 45 56, factor the expression and use the undamental identities to simplify. There is more than one orrect form of each answer. 5. tan x tan x sin x 46. sin x csc x sin x 7. sin x sec x sin x 48. cos x cos x tan x 9. sec x cos 50. x 4 sec x cos x. tan 4 x tan x 5. cos x cos 4 x 3. sin 4 x cos 4 x 54. sec 4 x tan 4 x 5. csc 3 x csc x csc x 6. sec 3 x sec x sec x n Exercises 57 60, perform the multiplication and use the undamental identities to simplify. There is more than one orrect form of each answer. 7. sin x cos x 8. cot x csc xcot x csc x 9. csc x csc x 0. 3 3 sin x3 3 sin x n Exercises 6 64, perform the addition or subtraction and se the fundamental identities to simplify. There is more han one correct form of each answer.. cos x cos x 6. 3. cos x sin x sin x cos x 64. n Exercises 65 68, rewrite the expression so that it is not in ractional form. There is more than one correct form of ach answer. 5. sin y cos y 66. 7. 3 sec x tan x 68. sec x sec x tan x sec x tan x 5 tan x sec x tan x csc x umerical and Graphical Analysis In Exercises 69 7, use graphing utility to complete the table and graph the unctions. Make a conjecture about and y. y 7. 7. In Exercises 73 76, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically. 73. cos x cot x sin x 74. 75. 76. y y sec 4 x sec x, sec x csc x tan x sin x cos x cos x cos x sin x, sin cos cos sin In Exercises 77 8, use the trigonometric substitution to write the algebraic expression as a trigonometric function of, where 0 < < /. 77. 9 x, x 3 cos 78. 64 6x, x cos 79. x 9, 80. x 4, 8. x 5, 8. x 00, x 3 sec x sec x 5 tan x 0 tan y In Exercises 83 86, use the trigonometric substitution to write the algebraic equation as a trigonometric function of, where / < < /. Then find sin and cos. 83. 3 9 x, x 3 sin 84. 3 36 x, x 6 sin 85. 6 4x, x cos 86. 53 00 x, x 0 cos In Exercises 87 90, use a graphing utility to solve the equation for, where 0 <. 87. sin cos 88. cos sin 89. sec tan 90. csc cot sin x cos x y tan x tan 4 x 9. 0. y cos x, y x 0. 0.4 0.6 0.8.0..4 y y y sin x sec x cos x, y sin x tan x In Exercises 9 94, rewrite the expression as a single logarithm and simplify the result. ln cos x ln sin x 9. 9. ln 93. ln sec x ln sin x cot t ln tan t 94. lncos t ln tan t
3330_050.qxd /5/05 9:5 AM Page 38 Section 5. Using Fundamental Identities 38 In Exercises 95 98, use a calculator to demonstrate the identity for each value of. 95. csc cot (a) 3, (b) 96. tan sec (a) (b) 97. cos sin (a) (b) 98. sin sin (a) (b) 346, 80, 50, 99. Friction The forces acting on an object weighing W units on an inclined plane positioned at an angle of with the horizontal (see figure) are modeled by W cos W sin 7 3. 0.8 where is the coefficient of friction. Solve the equation for and simplify the result. θ W In Exercises 03 06, fill in the blanks. (Note: The notation x c indicates that x approaches c from the right and x c indicates that x approaches c from the left.) 03. As x, sin x and csc x. 04. As x 0, cos x and sec x. 05. As x, tan x and cot x. 06. As and csc x. x, sin x In Exercises 07, determine whether or not the equation is an identity, and give a reason for your answer. 07. cos sin 08. cot csc 09. sin k cos k tan, k is a constant. 0.. 5 cos 5 sec sin csc. csc 3. Use the definitions of sine and cosine to derive the sin cos. 4. Writing Use the sin cos to derive the other Pythagorean identities, tan sec and cot csc. Discuss how to remember these identities and other fundamental identities. Skills Review 00. Rate of Change The rate of change of the function fx csc x sin x is given by the expression csc x cot x cos x. Show that this expression can also be written as cos x cot x. Synthesis True or False? In Exercises 0 and 0, determine whether the statement is true or false. Justify your answer. 0. The even and odd trigonometric identities are helpful for determining whether the value of a trigonometric function is positive or negative. 0. A cofunction identity can be used to transform a tangent function so that it can be represented by a cosecant function. In Exercises 5 and 6, perform the operation and simplify. 5. 6. x 5x 5 In Exercises 7 0, perform the addition or subtraction and simplify. 7. x 5 x x 8 8. 9. x x 4 7 x 4 0. In Exercises 4, sketch the graph of the function. (Include two full periods.). f x. sin x 4 z 3 6x x 4 3 4 x x x 5 x x 5 x f x tan 3. f x 4. f x 3 cosx 3 sec x