3330_0505.qxd 1/5/05 9:06 AM Page 407 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 407 5.5 Multiple Angle and Product-to-Sum Formulas What you should learn Use multiple-angle formulas to rewrite and evaluate trigonometric functions. Use power-reducing formulas to rewrite and evaluate trigonometric functions. Use half-angle formulas to rewrite and evaluate trigonometric functions. Use product-to-sum and sum-to-product formulas to rewrite and evaluate trigonometric functions. Use trigonometric formulas to rewrite real-life models. Why you should learn it You can use a variety of trigonometric formulas to rewrite trigonometric functions in more convenient forms. For instance, in Exercise 119 on page 417, you can use a double-angle formula to determine at what angle an athlete must throw a javelin. Mark Dadswell/Getty Images Consider having your students prove the double-angle formulas. Doubleangle formulas are often used in calculus. Multiple-Angle Formulas In this section, you will study four other categories of trigonometric identities. 1. The first category involves functions of multiple angles such as sin ku and cos ku.. The second category involves squares of trigonometric functions such as sin u. 3. The third category involves functions of half-angles such as sinu. 4. The fourth category involves products of trigonometric functions such as sin u cos v. You should learn the double-angle formulas because they are used often in trigonometry and calculus. For proofs of the formulas, see Proofs in Mathematics on page 45. Double-Angle Formulas sin u sin u cos u tan u Example 1 Solving a Multiple-Angle Equation Solve cos x sin x 0. Begin by rewriting the equation so that it involves functions of x rather than x. Then factor and solve as usual. cos x 0 and 1 sin x 0 x, 3 So, the general solution is tan u 1 tan u cos x sin x cos x 0 cos x sin x 0 cos x1 sin x 0 x 3 cos u cos u sin u cos u 1 1 sin u Write original equation. Double-angle formula Factor. Set factors equal to zero. s in 0, x n and x 3 n where n is an integer. Try verifying these solutions graphically. Now try Exercise 9.
3330_0505.qxd 1/5/05 9:06 AM Page 408 408 Chapter 5 Analytic Trigonometry Example Using Double-Angle Formulas to Analyze Graphs Use a double-angle formula to rewrite the equation y 4 cos x. Then sketch the graph of the equation over the interval 0,. y y = 4 cos x Using the double-angle formula for cos u, you can rewrite the original equation as y 4 cos x cos x 1 cos x. Write original equation. Factor. Use double-angle formula. 1 π π x Using the techniques discussed in Section 4.5, you can recognize that the graph of this function has an amplitude of and a period of. The key points in the interval 0, are as follows. 1 FIGURE 5.9 Maximum Intercept Minimum Intercept Maximum 0, 3, 4, 0, 4, 0 Two cycles of the graph are shown in Figure 5.9. Now try Exercise 1. Example 3 Evaluating Functions Involving Double Angles 4 y 4 6 8 10 1 4 6 13 (5, 1) x Use the following to find sin, cos, and tan. cos 5 13, < < From Figure 5.10, you can see that sin yr 113. Consequently, using each of the double-angle formulas, you can write sin cos tan 3 sin cos 1 1313 5 10 169 5 119 cos 1 1 169 169 sin 10 cos 119. FIGURE 5.10 Now try Exercise 3. The double-angle formulas are not restricted to angles and. Other double combinations, such as and or and 3, are also valid. Here are two examples. 4 6 sin 4 sin cos and cos 6 cos 3 sin 3 By using double-angle formulas together with the sum formulas given in the preceding section, you can form other multiple-angle formulas.
3330_0505.qxd 1/5/05 9:06 AM Page 409 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 409 Example 4 Deriving a Triple-Angle Formula sin 3x sinx x sin x cos x cos x sin x sin x cos x cos x 1 sin xsin x sin x cos x sin x sin 3 x sin x1 sin x sin x sin 3 x sin x sin 3 x sin x sin 3 x 3 sin x 4 sin 3 x Now try Exercise 97. Power-Reducing Formulas The double-angle formulas can be used to obtain the following power-reducing formulas. Example 5 shows a typical power reduction that is used in calculus. Power-reducing formulas are often used in calculus. Power-Reducing Formulas sin u 1 cos u cos u 1 cos u tan u 1 cos u 1 cos u For a proof of the power-reducing formulas, see Proofs in Mathematics on page 45. Example 5 Reducing a Power Rewrite sin 4 x as a sum of first powers of the cosines of multiple angles. Note the repeated use of power-reducing formulas. sin 4 x sin x 1 cos x Property of exponents Power-reducing formula 1 4 1 cos x cos x 1 1 cos 4x 1 cos x 4 Expand. Power-reducing formula 1 4 1 cos x 1 8 1 cos 4x 8 Distributive Property 1 3 4 cos x cos 4x 8 Factor out common factor. Now try Exercise 9.
3330_0505.qxd 1/5/05 9:06 AM Page 410 410 Chapter 5 Analytic Trigonometry Half-Angle Formulas You can derive some useful alternative forms of the power-reducing formulas by replacing u with u. The results are called half-angle formulas. Half-Angle Formulas sin u ± 1 cos u cos u ± 1 cos u tan u 1 cos u sin u sin u 1 cos u u The signs of sin u and cos u depend on the quadrant in which lies. Example 6 Using a Half-Angle Formula To find the exact value of a trigonometric function with an angle measure in form using a half-angle formula, first convert the angle measure to decimal degree form. Then multiply the resulting angle measure by. DMS Find the exact value of sin 105. Begin by noting that 105 is half of 10. Then, using the half-angle formula for sinu and the fact that 105 lies in Quadrant II, you have sin 105 1 cos 10 1 cos 30 1 3 3. The positive square root is chosen because sin is positive in Quadrant II. Now try Exercise 41. Use your calculator to verify the result obtained in Example 6. That is, evaluate sin 105 and 3. sin 105 0.965958 3 0.965958 You can see that both values are approximately 0.965958.
3330_0505.qxd 1/5/05 9:06 AM Page 411 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 411 Example 7 Solving a Trigonometric Equation Find all solutions of sin x cos x in the interval 0,. Algebraic sin x cos x sin x 1 cos x ± sin x 1 cos x sin x 1 cos x 1 cos x 1 cos x cos x cos x 0 cos xcos x 1 0 Write original equation. Half-angle formula Simplify. Simplify. Pythagorean identity Simplify. Factor. By setting the factors cos x and cos x 1 equal to zero, you find that the solutions in the interval 0, are x, x 3, and x 0. Now try Exercise 59. Graphical Use a graphing utility set in radian mode to graph y sin x cos x, as shown in Figure 5.11. Use the zero or root feature or the zoom and trace features to approximate the x-intercepts in the interval 0, to be x 0, x 1.571 and x 4.71 3,. These values are the approximate solutions of sin x cos x 0 in the interval 0,. 1 FIGURE 5.11 3 y = sin x cos x ( ) Product-to-Sum Formulas A common error is to write cos x as cos x, rather than to use the correct identity cos x 1 cos x. Each of the following product-to-sum formulas is easily verified using the sum and difference formulas discussed in the preceding section. Product-to-Sum Formulas sin u sin v 1 cosu v cosu v cos u cos v 1 cosu v cosu v sin u cos v 1 sinu v sinu v cos u sin v 1 sinu v sinu v Product-to-sum formulas are used in calculus to evaluate integrals involving the products of sines and cosines of two different angles.
3330_0505.qxd 1/5/05 9:06 AM Page 41 41 Chapter 5 Analytic Trigonometry Example 8 Writing Products as Sums Rewrite the product cos 5x sin 4x as a sum or difference. Using the appropriate product-to-sum formula, you obtain cos 5x sin 4x 1 sin5x 4x sin5x 4x 1 sin 9x 1 sin x. Now try Exercise 67. Occasionally, it is useful to reverse the procedure and write a sum of trigonometric functions as a product. This can be accomplished with the following sum-to-product formulas. Activities 1. Find the exact values of sin u, cos u, and tan u given cot u 5, 3 < u <. Answer: sin u 5 13 cos u 1 13 tan u 5 1. Find all solutions of the equation in the interval 0,. 4 sin x cos x 1 5 13 17 Answer: x 1, 1, 1, 1 3. Rewrite as a sum: sin sin. Answer: cos cos Sum-to-Product Formulas sin u sin v sin u v cos u v sin u sin v cos u v sin u v cos u cos v cos u v cos u v cos u cos v sin u v sin u v For a proof of the sum-to-product formulas, see Proofs in Mathematics on page 46. Example 9 Using a Sum-to-Product Formula Find the exact value of cos 195 cos 105. Using the appropriate sum-to-product formula, you obtain 195 105 cos 195 cos 105 cos cos 195 105 cos 150 cos 45 3 6. Now try Exercise 83.
3330_0505.qxd 1/7/05 4:13 PM Page 413 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 413 Example 10 Solving a Trigonometric Equation y 1 FIGURE 5.1 y = sin 5x + sin 3x 3π x Solve sin 5x sin 3x 0. Write original equation. Sum-to-product formula Simplify. By setting the factor sin 4x equal to zero, you can find that the solutions in the interval 0, are x 0, 4,, 3 4,, 5 4, 3, 7 4. The equation cos x 0 yields no additional solutions, and you can conclude that the solutions are of the form x n 4 sin 5x sin 3x 0 5x 3x sin cos 5x 3x 0 sin 4x cos x 0 where n is an integer. You can confirm this graphically by sketching the graph of y sin 5x sin 3x, as shown in Figure 5.1. From the graph you can see that the x-intercepts occur at multiples of 4. Now try Exercise 87. Example 11 Verifying a Trigonometric Identity Verify the identity sin t sin 3t tan t. cos t cos 3t Using appropriate sum-to-product formulas, you have t 3t sin sin t sin 3t cos t cos 3t cos t 3t t 3t cos cos t 3t sint cost cost cost sin t cos t tan t. Now try Exercise 105.
3330_0505.qxd 1/5/05 9:06 AM Page 414 414 Chapter 5 Analytic Trigonometry Application Example 1 Projectile Motion FIGURE 5.13 Not drawn to scale Ignoring air resistance, the range of a projectile fired at an angle with the horizontal and with an initial velocity of feet per second is given by r 1 16 v 0 sin cos where r is the horizontal distance (in feet) that the projectile will travel. A place kicker for a football team can kick a football from ground level with an initial velocity of 80 feet per second (see Figure 5.13). a. Write the projectile motion model in a simpler form. b. At what angle must the player kick the football so that the football travels 00 feet? c. For what angle is the horizontal distance the football travels a maximum? a. You can use a double-angle formula to rewrite the projectile motion model as r 1 3 v 0 sin cos 1 3 v 0 sin. Rewrite original projectile motion model. Rewrite model using a double-angle formula. b. r 1 Write projectile motion model. 3 v 0 sin v 0 00 1 3 80 sin Substitute 00 for r and 80 for v 0. 00 00 sin Simplify. 1 sin Divide each side by 00., 4. You know that so dividing this result by produces Because 4 45, you can conclude that the player must kick the football at an angle of 45 so that the football will travel 00 feet. c. From the model r 00 sin you can see that the amplitude is 00. So the maximum range is r 00 feet. From part (b), you know that this corresponds to an angle of 45. Therefore, kicking the football at an angle of 45 will produce a maximum horizontal distance of 00 feet. Now try Exercise 119. W RITING ABOUT MATHEMATICS Deriving an Area Formula Describe how you can use a double-angle formula or a half-angle formula to derive a formula for the area of an isosceles triangle. Use a labeled sketch to illustrate your derivation. Then write two examples that show how your formula can be used.
3330_0505.qxd 1/5/05 9:06 AM Page 415 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 415 5.5 Exercises VOCABULARY CHECK: 1. sin u. 3. cos u 4. Fill in the blank to complete the trigonometric formula. 1 cos u 1 cos u 1 cos u 5. sin u 6. tan u 7. cos u cos v 8. sin u cos v 9. sin u sin v 10. cos u cos v PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.eduspace.com. In Exercises 1 8, use the figure to find the exact value of the trigonometric function. 4 1. sin. tan 3. cos 4. sin 5. tan 6. sec 7. csc 8. cot In Exercises 9 18, find the exact solutions of the equation in the interval [0,. 9. sin x sin x 0 10. sin x cos x 0 11. 4 sin x cos x 1 1. sin x sin x cos x 13. cos x cos x 0 14. cos x sin x 0 15. tan x cot x 0 16. tan x cos x 0 17. sin 4x sin x 18. sin x cos x 1 1 5. tan u 3 4, 6. cot u 4, 7. sec u 5, 8. csc u 3, 0 < u < 3 < u < < u < < u < In Exercises 9 34, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. 9. cos 4 x 30. sin 8 x 31. sin x cos x 3. sin 4 x cos 4 x 33. sin x cos 4 x 34. sin 4 x cos x In Exercises 35 40, use the figure to find the exact value of the trigonometric function. In Exercises 19, use a double-angle formula to rewrite the expression. 19. 6 sin x cos x 0. 6 cos x 3 1. 4 8 sin x. cos x sin xcos x sin x In Exercises 3 8, find the exact values of sin u, cos u, and tan u using the double-angle formulas. 3. sin u 4 5, 4. cos u 3, < u < 3 < u < 35. cos 36. sin 37. tan 38. sec 15 39. csc 40. cot 8
3330_0505.qxd 1/5/05 9:06 AM Page 416 416 Chapter 5 Analytic Trigonometry In Exercises 41 48, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. 41. 75 4. 165 43. 11 30 44. 67 30 In Exercises 49 54, find the exact values of sinu/, cosu/, and tanu/ using the half-angle formulas. 49. 50. cos u 3 5, 51. tan u 5 8, 5. 45. 46. 8 1 3 sin u 5 13, cot u 3, 53. csc u 5 3, 54. sec u 7, In Exercises 55 58, use the half-angle formulas to simplify the expression. 55. 1 cos 6x 56. 57. 1 cos 8x 1 cos 8x 58. In Exercises 59 6, find all solutions of the equation in the interval [0,. Use a graphing utility to graph the equation and verify the solutions. 59. sin x cos x 0 60. sin x cos x 1 0 61. cos x sin x 0 6. tan x sin x 0 In Exercises 63 74, use the product-to-sum formulas to write the product as a sum or difference. 5 63. 6 sin 64. 4 cos sin 4 cos 4 3 6 65. 10 cos 75 cos 15 66. 6 sin 45 cos 15 67. cos 4 sin 6 68. 3 sin sin 3 69. 5 cos5 cos 3 70. cos cos 4 < u < 0 < u < 3 < u < 3 < u < 3 < u < < u < 7 47. 48. 8 1 1 cos 4x 1 cosx 1 71. sinx y sinx y 7. sinx y cosx y 73. 74. cos sin In Exercises 75 8, use the sum-to-product formulas to write the sum or difference as a product. 75. sin 5 sin 3 76. sin 3 sin 77. cos 6x cos x 78. sin x sin 5x 79. 80. 81. cos cos 8. sin x sin sin cos cos sin x In Exercises 83 86, use the sum-to-product formulas to find the exact value of the expression. 83. sin 60 sin 30 84. cos 10 cos 30 85. 86. sin 5 3 cos 3 cos sin 4 4 4 4 In Exercises 87 90, find all solutions of the equation in the interval [0,. Use a graphing utility to graph the equation and verify the solutions. 87. sin 6x sin x 0 88. cos x cos 6x 0 cos x 89. 1 0 90. sin 3x sin x 0 sin 3x sin x In Exercises 91 94, use the figure and trigonometric identities to find the exact value of the trigonometric function in two ways. β 4 α 91. sin 9. cos 93. sin cos 94. cos sin In Exercises 95 110, verify the identity. 95. csc csc cos 96. sec sec sec 97. cos sin cos 4 98. cos 4 x sin 4 x cos x 99. sin x cos x 1 sin x 3 1 sin sin 5
3330_0505.qxd 1/5/05 9:06 AM Page 417 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 417 100. 101. 10. 103. 104. sin 3 cos 3 1 sin 3 1 cos 10y cos 5y cos 3 1 4 sin cos sec u ± tan u tan u sin u tan u csc u cot u 10. Geometry The length of each of the two equal sides of an isosceles triangle is 10 meters (see figure). The angle between the two sides is. 10 m 10 m sin x ± sin y x ± y 105. tan cos x cos y sin x sin y x y 106. cot cos x cos y cos 4x cos x 107. cot 3x sin 4x sin x cos t cos 3t 108. cot t sin 3t sin t 109. sin 6 x sin 6 x cos x 110. cos In Exercises 111 114, use a graphing utility to verify the identity. Confirm that it is an identity algebraically. 111. cos 3 cos 3 3 sin cos 11. sin 4 4 sin cos 1 sin 113. cos 4x cos x sin 3x sin x 114. cos 3x cos xsin 3x sin x tan x In Exercises 115 and 116, graph the function by hand in the interval [0, ] by using the power-reducing formulas. 115. f x sin x 116. f x) cos x In Exercises 117 and 118, write the trigonometric expression as an algebraic expression. 117. sin arcsin x 118. cos arccos x 119. Projectile Motion The range of a projectile fired at an angle with the horizontal and with an initial velocity of feet per second is v 0 3 x cos 3 x cos x (a) Write the area of the triangle as a function of. (b) Write the area of the triangle as a function of. Determine the value of such that the area is a maximum. 11. Mach Number The mach number M of an airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane (see figure). The mach number is related to the apex angle of the cone by sin 1 M. (a) Find the angle that corresponds to a mach number of 1. (b) Find the angle that corresponds to a mach number of 4.5. (c) The speed of sound is about 760 miles per hour. Determine the speed of an object with the mach numbers from parts (a) and (b). (d) Rewrite the equation in terms of. Model It r 1 3 v 0 sin where r is measured in feet. An athlete throws a javelin at 75 feet per second. At what angle must the athlete throw the javelin so that the javelin travels 130 feet?
3330_0505.qxd 1/5/05 9:06 AM Page 418 418 Chapter 5 Analytic Trigonometry 1. Railroad Track When two railroad tracks merge, the overlapping portions of the tracks are in the shapes of circular arcs (see figure). The radius of each arc r (in feet) and the angle are related by Write a formula for x in terms of cos. Synthesis True or False? In Exercises 13 and 14, determine whether the statement is true or false. Justify your answer. 13. Because the sine function is an odd function, for a negative number u, sin u sin u cos u. 14. sin u when u is in the second quadrant. 1 cos u In Exercises 15 and 16, (a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval [0, and (b) solve the trigonometric equation and verify that its solutions are the x-coordinates of the maximum and minimum points of f. (Calculus is required to find the trigonometric equation.) 15. 16. x r sin. Function f x 4 sin x cos x f xcos x sin x Trigonometric Equation 17. Exploration Consider the function given by f x sin 4 x cos 4 x. r x cos x sin x 0 cos x sin x 1 0 (a) Use the power-reducing formulas to write the function in terms of cosine to the first power. (b) Determine another way of rewriting the function. Use a graphing utility to rule out incorrectly rewritten functions. (c) Add a trigonometric term to the function so that it becomes a perfect square trinomial. Rewrite the function as a perfect square trinomial minus the term that you added. Use a graphing utility to rule out incorrectly rewritten functions. r (d) Rewrite the result of part (c) in terms of the sine of a double angle. Use a graphing utility to rule out incorrectly rewritten functions. (e) When you rewrite a trigonometric expression, the result may not be the same as a friend s. Does this mean that one of you is wrong? Explain. 18. Conjecture Consider the function given by f x sin x cos x 1. (a) Use a graphing utility to graph the function. (b) Make a conjecture about the function that is an identity with f. (c) Verify your conjecture analytically. Skills Review In Exercises 19 13, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment connecting the points. 19. 5,, 1, 4 130. 131. 13. 4, 3, 6, 10 0, 1, 4 3, 5 1 3, 3, 1, 3 In Exercises 133 136, find (if possible) the complement and supplement of each angle. 133. (a) 55 (b) 16 134. (a) 109 (b) 78 9 135. (a) (b) 18 0 136. (a) 0.95 (b).76 137. Profit The total profit for a car manufacturer in October was 16% higher than it was in September. The total profit for the months was $507,600. Find the profit for each month. 138. Mixture Problem A 55-gallon barrel contains a mixture with a concentration of 30%. How much of this mixture must be withdrawn and replaced by 100% concentrate to bring the mixture up to 50% concentration? 139. Distance A baseball diamond has the shape of a square in which the distance between each of the consecutive bases is 90 feet. Approximate the straight-line distance from home plate to second base.