Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Study Guide and Review - Chapter 5 Complete each identity by filling in the blank. Then name the identity. = _______ 1. sec SOLUTION: = The reciprocal identity states sec . 2. _______ = SOLUTION: The quotient identity states tan = . 3. _______ + 1 = sec2 SOLUTION: The Pythagorean identity states sec 4. cos (90° – = . ) = _______ SOLUTION: The cofunction identity states cos(90° − ) = sin . 5. tan (– ) = _______ SOLUTION: The odd-even identity states tan(− ) = −tan 6. sin ( + ) = sin _______ + cos . _______ SOLUTION: The sum identity states sin (α + β) = sin α cos β + cos α sin β. 7. _______ = cos2α – sin2 α SOLUTION: 2 2 The double-angle identity states that cos 2α = cos α – sin α. 8. _______= ± SOLUTION: The half-angle identity states cos 9. = ± . =_______ SOLUTION: 2 = sin The power-reducing identity states eSolutions Manual - Powered by Cognero 10. _______ = [cos ( – . Page 1 ) + cos ( + )] SOLUTION: Study and Review - Chapter 5 TheGuide half-angle identity states cos = ± 9. . =_______ SOLUTION: 2 = sin The power-reducing identity states 10. _______ = [cos ( – ) + cos ( + . )] SOLUTION: The product-to-sum identity states cos α cos β = [cos (α – β) + cos (α + β)]. Find the value of each expression using the given information. and cos ; tan = 3, cos > 0 11. sec SOLUTION: Use the Pythagorean Identity that involves tan Since we are given that cos to find sec . is positive, sec must be positive. So, sec = . Use the reciprocal identity to find cos . 12. cot and sin ; cos = , tan < 0 SOLUTION: Use the Pythagorean Identity that involves cos eSolutions Manual - Powered by Cognero to find sin . Page 2 Study Guide and Review - Chapter 5 12. cot and sin ; cos = , tan < 0 SOLUTION: Use the Pythagorean Identity that involves cos Since tan = quotient identity cot 13. csc and tan ; cos to find sin . is negative and cos is negative, sin must be positive. So, sin = = . Use the to find cot . = , sin < 0 SOLUTION: Use the Pythagorean Identity that involves cos Since sin < 0, to find sin . . Use the reciprocal identity csc eSolutions Manual - Powered by Cognero = to find csc . Page 3 Study Guide and Review - Chapter 5 13. csc and tan ; cos = , sin < 0 SOLUTION: Use the Pythagorean Identity that involves cos Since sin < 0, and cos ; tan = . Use the reciprocal identity csc Use the quotient identity tan 14. cot to find sin . = = to find csc . to find tan . , csc > 0 SOLUTION: Use the reciprocal function cot eSolutions Manual - Powered by Cognero = to find cot . Use the Pythagorean identity that involves tan to find sec . Page 4 Study Guide and Review - Chapter 5 14. cot and cos ; tan = , csc > 0 SOLUTION: Use the reciprocal function cot = to find cot . Use the Pythagorean identity that involves tan Since csc is positive, sin positive. Since cos to find sec is positive. Since tan = . is positive and sin is positive, cos has to be is positive, sec is positive. So, sec = . Use the reciprocal identity cos = to find cos . 15. sec and sin ; cot = –2, csc < 0 SOLUTION: Use the Pythagorean identity that involves cot Since csc - Powered < 0, csc = − eSolutions Manual by Cognero to find csc . . Use the reciprocal identity sin = to find sin . Page 5 Study Guide and Review - Chapter 5 15. sec and sin ; cot = –2, csc < 0 SOLUTION: Use the Pythagorean identity that involves cot Since csc < 0, csc = − . Use the reciprocal identity sin Use the Pythagorean identity that involves sin Since cot = reciprocal identity sec 16. cos and sin ; cot to find csc . = to find sin . to find cos . is negative and sin is negative, cos must be positive. So, cos = to find sec . = = . Use the , sec < 0 eSolutions Manual - Powered by Cognero Page 6 SOLUTION: Use the Pythagorean identity that involves cot to find csc . Study Guide and Review - Chapter 5 16. cos and sin ; cot = , sec < 0 SOLUTION: Use the Pythagorean identity that involves cot to find csc Since sec sin < 0, cos is negative. Since cot is negative, csc must be negative. So, = . is positive and cos is negative, sin is negative. Since . Use the reciprocal identity sin = to find sin . Use the Pythagorean identity that involves sin It was determined that cos to find cos . is negative. So, Simplify each expression. 17. sin2 (–x) + cos2(–x) SOLUTION: eSolutions Manual - Powered by Cognero Page 7 Study Guide and that Review - Chapter 5 It was determined cos is negative. So, Simplify each expression. 17. sin2 (–x) + cos2(–x) SOLUTION: 18. sin2 x + cos2 x + cot2 x SOLUTION: 19. SOLUTION: 20. SOLUTION: 21. SOLUTION: eSolutions Manual - Powered by Cognero Page 8 SOLUTION: Study Guide and Review - Chapter 5 21. SOLUTION: 22. SOLUTION: Verify each identity. 23. + = 2 csc SOLUTION: eSolutions Manual - Powered by Cognero Page 9 Study Guide and Review - Chapter 5 Verify each identity. 23. = 2 csc + SOLUTION: 24. = 1 + SOLUTION: 25. + = 2 sec SOLUTION: eSolutions Manual - Powered by Cognero Page 10 Study Guide and Review - Chapter 5 25. = 2 sec + SOLUTION: 26. = SOLUTION: 27. = csc – 1 SOLUTION: eSolutions Manual - Powered by Cognero Page 11 Study Guide and Review - Chapter 5 27. = csc – 1 SOLUTION: 28. + = sec + csc SOLUTION: 29. = csc SOLUTION: eSolutions Manual - Powered by Cognero Page 12 Study Guide and Review - Chapter 5 29. = csc SOLUTION: 30. cot 2 csc + sec = csc sec SOLUTION: 31. = SOLUTION: eSolutions Manual - Powered by Cognero Page 13 Study Guide and Review - Chapter 5 31. = SOLUTION: 32. cos4 4 – sin = SOLUTION: Find all solutions of each equation on the interval [0, 2π]. 33. 2 sin x = SOLUTION: On the interval [0, 2π), when x = and x = . 34. 4 cos2 x = 3 SOLUTION: eSolutions Manual - Powered by Cognero Page 14 On the interval [0, 2π), when x = Study Guide and Review - Chapter 5 and x = . 34. 4 cos2 x = 3 SOLUTION: On the interval [0, 2 ), when x = and x = and when x = and x = . 35. tan2 x – 3 = 0 SOLUTION: 2 tan x – 3 = 0 On the interval [0, 2π), when x = and x = and when x = and x = . when x = and x = and when x = and x = . 36. 9 + cot2 x = 12 SOLUTION: On the interval [0, 2π), 37. 2 sin2 x = sin x SOLUTION: On the interval [0, 2π), sin x = 0 when x = 0 and 38. 3 cos x + 3 = sin2 x eSolutions Manual - Powered by Cognero SOLUTION: and when x = and x = . Page 15 Study Guide and Review On the interval [0, 2π), sin x =-0Chapter when x = 0 5and and when x = and x = . 38. 3 cos x + 3 = sin2 x SOLUTION: On the interval [0, 2 ), cos x = −1 when x = the cosine function can attain is −1. . The equation cos x = −2 has no solution since the minimum value Solve each equation for all values of x. 39. sin2 x – sin x = 0 SOLUTION: The period of sine is 2 , so you only need to find solutions on the interval interval are 0 and and the solution to sin x = 1 on this interval is . . The solutions to sin x = 0 on this Solutions on the interval (– , ), are found by adding integer multiples of 2π. The solutions x = 0 + 2n and x = + 2n can be combined to x = n . Therefore, the general form of the solutions is n , + 2n , where n is an integer. 40. tan2 x = tan x SOLUTION: eSolutions Manual - Powered by Cognero The period of tangent is , so you only need to find solutions on the interval Page 16 . The solution to tan x = 0 on this Solutions on the interval (– , ), are found by adding integer multiples of 2π. The solutions x = 0 + 2n and x = + 2n can be combined to x = n . Therefore, the general form of the solutions is n , + 2n , where n is an Study Guide and Review - Chapter 5 integer. 40. tan2 x = tan x SOLUTION: The period of tangent is , so you only need to find solutions on the interval interval is 0 and the solution to tan x = 1 on this interval is . . The solution to tan x = 0 on this Solutions on the interval (– , ) are found by adding integer multiples of π. Therefore, the general form of the solutions is nπ, + n , where n is an integer. 41. 3 cos x = cos x – 1 SOLUTION: The period of cosine is 2 , so you only need to find solutions on the interval on this interval are Solutions on the interval (– + 2n , solutions is , and . The solutions to . ) are found by adding integer multiples of 2π. Therefore, the general form of the + 2n , where n is an integer. 42. sin2 x = sin x + 2 SOLUTION: The period of sine is 2 , so you only need to find solutions on the interval . The equation sin x = 2 has no solution since the maximum value the sine function can attain is 1. The solution to sin x = −1 on this interval is . eSolutions Manual - Powered by Cognero Page 17 Solutions on the interval (– , ) are found by adding integer multiples of 2 . Therefore, the general form of the solutions is + 2n , where n is an integer. on this interval are Solutions on the interval (– , and . ) are found by adding integer multiples of 2π. Therefore, the general form of the Study Guide Review - Chapter solutions is and + 2n , + 2n , where n5is an integer. 42. sin2 x = sin x + 2 SOLUTION: The period of sine is 2 , so you only need to find solutions on the interval . The equation sin x = 2 has no solution since the maximum value the sine function can attain is 1. The solution to sin x = −1 on this interval is . Solutions on the interval (– , ) are found by adding integer multiples of 2 . Therefore, the general form of the solutions is + 2n , where n is an integer. 43. sin2 x = 1 – cos x SOLUTION: The period of cosine is 2 , so you only need to find solutions on the interval this interval are and . The solutions to cos x = 0 on and the solution to cos x = 1 on this interval is 0. Solutions on the interval (– , ), are found by adding integer multiples of 2π. The solutions x = + 2n and x = + 2n can be combined to x = Therefore, the general form of the solutions is 2n , + n . + n , where n is an integer. 44. sin x = cos x + 1 SOLUTION: eSolutions Manual - Powered by Cognero Page 18 The solutions x = + 2n and x = + 2n can be combined to x = + n . Study Guidetheand Review Chapter 5 2n Therefore, general form of-the solutions is , + n , where n is an integer. 44. sin x = cos x + 1 SOLUTION: The period of cosine is 2 , so you only need to find solutions on the interval this interval are and and the solution to cos x = −1 on this interval is . The solutions to cos x = 0 on . Since each side of the equation was squared, check for extraneous solutions. Solutions on the interval (– , ), are found by adding integer multiples of 2 . Therefore, the general form of the solutions is + 2n , + 2n , where n is an integer. Find the exact value of each trigonometric expression. 45. cos 15 SOLUTION: Write 15° as the sum or difference of angle measures with cosines that you know. eSolutions Manual - Powered by Cognero Page 19 Solutions on the interval (– , ), are found by adding integer multiples of 2 . Therefore, the general form of the Study Guide 5 integer. solutions is and + 2n Review , + 2n - ,Chapter where n is an Find the exact value of each trigonometric expression. 45. cos 15 SOLUTION: Write 15° as the sum or difference of angle measures with cosines that you know. 46. sin 345 SOLUTION: Write 345° as the sum or difference of angle measures with sines that you know. 47. tan SOLUTION: eSolutions Manual - Powered by Cognero Page 20 Study Guide and Review - Chapter 5 47. tan SOLUTION: 48. sin SOLUTION: Write as the sum or difference of angle measures with sines that you know. eSolutions Manual - Powered by Cognero 49. cos Page 21 Study Guide and Review - Chapter 5 49. cos SOLUTION: Write as the sum or difference of angle measures with cosines that you know. 50. tan SOLUTION: Write as the sum or difference of angle measures with tangents that you know. Simplify each expression. eSolutions Manual - Powered by Cognero 51. Page 22 Study Guide and Review - Chapter 5 Simplify each expression. 51. SOLUTION: 52. cos 24 cos 36 − sin 24 sin 36 SOLUTION: 53. sin 95 cos 50 − cos 95 sin 50 SOLUTION: 54. cos cos + sin sin SOLUTION: Verify each identity. 55. cos ( + 30 ) – sin ( + 60 ) = –sin SOLUTION: eSolutions Manual - Powered by Cognero Page 23 Study Guide and Review - Chapter 5 Verify each identity. 55. cos ( + 30 ) – sin ( + 60 ) = –sin SOLUTION: 56. SOLUTION: 57. SOLUTION: 58. SOLUTION: eSolutions Manual - Powered by Cognero Page 24 Study Guide and Review - Chapter 5 58. SOLUTION: Find the values of sin 2 , cos 2 , and tan 2 59. cos = for the given value and interval. , (0 , 90 ) SOLUTION: Since on the interval (0°, 90°), one point on the terminal side of θ has x-coordinate 1 and a distance of 3 units from the origin as shown. The y-coordinate of this point is therefore Using this point, we find that and or 2 . Now use the double-angle identities for sine, cosine, and tangent to find sin 2 , cos 2 , and tan 2 . eSolutions Manual - Powered by Cognero Page 25 Study Guide and Review - Chapter 5 Find the values of sin 2 , cos 2 , and tan 2 59. cos = for the given value and interval. , (0 , 90 ) SOLUTION: on the interval (0°, 90°), one point on the terminal side of θ has x-coordinate 1 and a distance of 3 Since units from the origin as shown. The y-coordinate of this point is therefore and Using this point, we find that or 2 . Now use the double-angle identities for sine, cosine, and tangent to find sin 2 , cos 2 , and tan 2 . 60. tan = 2, (180 , 270 ) eSolutions Manual - Powered by Cognero Page 26 SOLUTION: If tan = 2, then tan = . Since tan = on the interval (180°, 270°), one point on the terminal side of has Study Guide and Review - Chapter 5 60. tan = 2, (180 , 270 ) SOLUTION: If tan = 2, then tan = . Since tan = on the interval (180°, 270°), one point on the terminal side of has x-coordinate −1 and y-coordinate −2 as shown. The distance from the point to the origin is Using this point, we find that and or . . Now use the double-angle identities for sine, cosine, and tangent to find sin 2 , cos 2 , and tan 2 . 61. SOLUTION: eSolutions Manual - Powered by Cognero Since on the interval Page 27 , one point on the terminal side of θ has y-coordinate 4 and a distance of 5 Study Guide and Review - Chapter 5 61. SOLUTION: Since on the interval , one point on the terminal side of θ has y-coordinate 4 and a distance of 5 units from the origin as shown. The x-coordinate of this point is therefore − or −3. Using this point, we find that and Now use the double-angle identities for sine and cosine to find sin 2θ and cos 2θ. Use the definition of tangent to find tan 2θ. eSolutions 62. Manual - Powered by Cognero SOLUTION: Page 28 Study Guide and Review - Chapter 5 62. SOLUTION: If , then . Since on the interval , one point on the terminal side of θ has x- coordinate 5 and a distance of 13 units from the origin as shown. The y-coordinate of this point is therefore – or –12. Using this point, we find that and Now use the double-angle identities for sine and cosine to find sin 2θ and cos 2θ. Use the definition of tangent to find tan 2θ. Find the exact value of each expression. 63. sin 75 eSolutions Manual - Powered by Cognero SOLUTION: Page 29 Notice that 75° is half of 150°. Therefore, apply the half-angle identity for sine, noting that since 75° lies in Quadrant I, its sine is positive. Study Guide and Review - Chapter 5 Find the exact value of each expression. 63. sin 75 SOLUTION: Notice that 75° is half of 150°. Therefore, apply the half-angle identity for sine, noting that since 75° lies in Quadrant I, its sine is positive. 64. cos SOLUTION: Notice that is half of . Therefore, apply the half-angle identity for cosine, noting that since lies in Quadrant II, its cosine is negative. 65. tan 67.5 SOLUTION: Notice that 67.5° is half of 135°. Therefore, apply the half-angle identity for tangent, noting that since 67.5° lies in Quadrant I, its tangent is positive. eSolutions Manual - Powered by Cognero Page 30 Study Guide and Review - Chapter 5 65. tan 67.5 SOLUTION: Notice that 67.5° is half of 135°. Therefore, apply the half-angle identity for tangent, noting that since 67.5° lies in Quadrant I, its tangent is positive. 66. cos SOLUTION: Notice that is half of . Therefore, apply the half-angle identity for cosine, noting that since lies in Quadrant I, its cosine is positive. 67. sin SOLUTION: Notice that is half of . Therefore, apply the half-angle identity for sine, noting that since lies in Quadrant IV, its sine is negative. eSolutions Manual - Powered by Cognero Page 31 Study Guide and Review - Chapter 5 67. sin SOLUTION: Notice that is half of . Therefore, apply the half-angle identity for sine, noting that since lies in Quadrant IV, its sine is negative. 68. tan SOLUTION: is half of Notice that . Therefore, apply the half-angle identity for tangent, noting that since lies in Quadrant III, its tangent is positive. 69. CONSTRUCTION Find the tangent of the angle that the ramp makes with the building if sin = = and cos . eSolutions Manual - Powered by Cognero SOLUTION: Page 32 Study Guide and Review - Chapter 5 69. CONSTRUCTION Find the tangent of the angle that the ramp makes with the building if sin = = and cos . SOLUTION: Use the Quotient Identity tan θ = Therefore, the tangent of the angle the ramp makes with the building is . 70. LIGHT The intensity of light that emerges from a system of two polarizing lenses can be calculated by I = I0 − , where I0 is the intensity of light entering the system and is the angle of the axis of the second lens with the first lens. Write the equation for the light intensity using only tan . SOLUTION: eSolutions Manual - Powered by Cognero Page 33 Study Guidetheand Review - Chapter Therefore, tangent of the angle the ramp 5 makes with the building is . 70. LIGHT The intensity of light that emerges from a system of two polarizing lenses can be calculated by I = I0 − , where I0 is the intensity of light entering the system and is the angle of the axis of the second lens with the first lens. Write the equation for the light intensity using only tan . SOLUTION: 71. MAP PROJECTIONS Stereographic projection is used to project the contours of a three-dimensional sphere onto a two-dimensional map. Points on the sphere are related to points on the map using r = . Verify that r = . SOLUTION: Verify that = eSolutions Manual - Powered by Cognero . Page 34 Study Guide and Review - Chapter 5 71. MAP PROJECTIONS Stereographic projection is used to project the contours of a three-dimensional sphere onto a two-dimensional map. Points on the sphere are related to points on the map using r = . Verify that r = . SOLUTION: Verify that = . By substitution, r = . 72. PROJECTILE MOTION A ball thrown with an initial speed of v0 at an angle d will remain in the air t seconds, where t = that travels a horizontal distance . Suppose a ball is thrown with an initial speed of 50 feet per second, travels 100 feet, and is in the air for 4 seconds. Find the angle at which the ball was thrown. SOLUTION: Let t = 4, d = 100, v0 = 50, and solve for θ. eSolutions Manual - Powered by Cognero Page 35 On the unit circle, cos θ = when θ = 60º and θ = 300º. Because the inverse cosine function is restricted to acute Study Guide and 5 By substitution, r =Review - Chapter . 72. PROJECTILE MOTION A ball thrown with an initial speed of v0 at an angle d will remain in the air t seconds, where t = that travels a horizontal distance . Suppose a ball is thrown with an initial speed of 50 feet per second, travels 100 feet, and is in the air for 4 seconds. Find the angle at which the ball was thrown. SOLUTION: Let t = 4, d = 100, v0 = 50, and solve for θ. On the unit circle, cos θ = when θ = 60º and θ = 300º. Because the inverse cosine function is restricted to acute angles of θ on the interval [0, 180º], θ = 60º. Therefore, the ball was thrown at an angle of 60º. 73. BROADCASTING Interference occurs when two waves pass through the same space at the same time. It is destructive if the amplitude of the sum of the waves is less than the amplitudes of the individual waves. Determine whether the interference is destructive when signals modeled by y = 20 sin (3t + 45 ) and y = 20 sin (3t + 225 ) are combined. SOLUTION: The sum of the two functions is zero, which means that the amplitude is 0. Because the amplitude of each of the original functions was 20 and 0 < 20, the combination of the two waves can be characterized as producing destructive interference. 74. TRIANGULATION Triangulation is the process of measuring a distance d using the angles α and β and the distance ℓ using . a. Solve the formulabyfor d. eSolutions Manual - Powered Cognero b. Verify that d = Page 36 . distance ℓ using . Study Guide and Review - Chapter 5 a. Solve the formula for d. b. Verify that d = c. Verify that d = d. Show that if = . . , then d = 0.5ℓ tan SOLUTION: a. b. c. d. eSolutions Manual - Powered by Cognero Page 37 Study Guide and Review - Chapter 5 d. eSolutions Manual - Powered by Cognero Page 38