Δ RST cap delta has a right angle at T. Use identities to show that each equation is true.

34. sin 2 R = 2 r s t 2 sine 2 r equals . fraction 2 r s , over t squared end fraction
35. sin 2 R = s 2 − r 2 t 2 sine 2 r equals . fraction s squared , minus , r squared , over t squared end fraction
36. sin 2 S = sin 2 R sine 2 s equals sine 2 r
37. sin 2 S 2 = t − r 2 t sine squared . s over 2 , equals . fraction t minus r , over 2 t end fraction
38. tan R 2 = r t + s tangent , r over 2 , equals . fraction r , over t plus s end fraction
39. tan 2 S 2 = t − r t + r tangent squared . s over 2 , equals . fraction t minus r , over t plus r end fraction
40. Reasoning If sin 2 A = sin 2 B , sine 2 eh equals sine 2 b comma must A = B ? eh equals b question mark Explain.

Given cos θ = 3 5 cosine theta equals , 3 fifths and 270 ° < θ < 360 ° , 270 degrees less than theta less than 360 degrees comma find the exact value of each expression.

41. sin   2 θ sine 2 theta
42. cos 2 θ cosine 2 theta
43. tan 2 θ 2 theta
44. csc 2 θ 2 theta
45. sin θ 2 sine , theta over 2
46. cos θ 2 cosine , theta over 2
47. tan θ 2 tangent , theta over 2
48. cot θ 2 co-tangent , theta over 2

Use identities to write each equation in terms of the single angle θ . theta . Then solve the equation for 0 ≤ θ < 2 π . 0 less than or equal to theta less than 2 pi .

49. 4 sin 2 θ − 3 cos θ = 0 4 sine 2 theta negative 3 cosine theta equals 0
50. 2 sin 2 θ − 3 sin θ = 0 2 sine 2 theta negative 3 sine theta equals 0
51. sin 2 θ sin θ = cos θ sine 2 theta sine theta equals cosine theta
52. cos 2 θ = − 2 cos 2 θ cosine 2 theta equals negative 2 cosine 2 theta

Simplify each expression.

53. 2 cos 2 θ − cos 2 θ 2 , cosine squared , theta negative cosine 2 theta
54. sin 2 θ 2 − cos 2 θ 2 sine squared . theta over 2 , minus , cosine squared . theta over 2
55. cos 2 θ sin θ + cos θ fraction cosine 2 theta , over sine theta plus cosine theta end fraction
56. 1. Write an identity for sin 2 θ sine squared , theta by using the double-angle identity cos 2 θ = 1 − 2 sin 2 θ . cosine 2 theta equals 1 minus 2 , sine squared , theta . The resulting identity is called a power reduction identity.
    2. Find a power reduction identity for cos 2 θ cosine squared , theta using a double-angle identity.
57. Open-Ended Choose an angle measure A.
    1. Find sin A and cos A.
    2. Use an identity to find sin 2A.
    3. Use an identity to find cos A 2 . cosine , eh over 2 , .
58. Writing Consider the graph of y = 1 − cos A 1 + cos A . y equals . square root of fraction 1 minus cosine eh , over 1 plus cosine eh end fraction end root . . Describe the period and any asymptotes if they exist.

C Challenge

Use double-angle identities to write each expression, using trigonometric functions of θ theta instead of 4 θ . 4 theta .

59. sin 4 θ sine 4 theta
60. cos 4 θ cosine 4 theta
61. tan 4 θ tangent 4 theta

Use half-angle identities to write each expression, using trigonometric functions of θ theta instead of θ 4 . theta over 4 , .

62. sin θ 4 sine , theta over 4
63. cos θ 4 cosine , theta over 4
64. tan θ 4 tangent , theta over 4
65. Use the Tangent Half-Angle Identity and a Pythagorean identity to prove each identity.
    1. tan A 2 = sin A 1 + cos A tangent , eh over 2 , equals . fraction sine eh , over 1 plus cosine eh end fraction
    2. tan A 2 = 1 − cos A sin A tangent , eh over 2 , equals . fraction 1 minus cosine eh , over sine eh end fraction

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