Ch. 3 - Trigonometric Identities and Equations
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- In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. sin 195°
Problem 15
- In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2 sin 15° cos 15°
Problem 15
- Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin 75°
Problem 16
- In Exercises 1–60, verify each identity. cos² θ (1 + tan² θ) = 1
Problem 16
- In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. 5𝝅 tan --------- 12
Problem 17
- In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. cos² 105° - sin² 105°
Problem 18
- In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 𝝅 2 cos² ------ ﹣ 1 8
Problem 19
- Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. ( 𝝅 𝝅 ) tan ( ------ + ------ ) ( 3 4 )
Problem 22
- In Exercises 1–60, verify each identity. cot² t ------------ = csc t - sin t csc t
Problem 22
- Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. ( 5𝝅 𝝅 ) tan ( ------ ﹣ ------) ( 3 4 )
Problem 24
- Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. sin 40° cos 20° + cos 40° sin 20°
Problem 26
- Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. 5𝝅 𝝅 5𝝅 𝝅 sin ------- cos -------- ﹣ cos -------- sin ------- 12 4 12 4
Problem 29
- In Exercises 35–38, find the exact value of the following under the given conditions: a. sin(α + β) 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2
Problem 35
- In Exercises 35–38, find the exact value of the following under the given conditions: c. tan(α + β) 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2
Problem 35
- In Exercises 35–38, find the exact value of the following under the given conditions: d. sin 2α 3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2
Problem 35
- In Exercises 35–38, find the exact value of the following under the given conditions: a. sin(α + β) 1 3𝝅 1 3𝝅 sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 3 2 3 2
Problem 38
- In Exercises 35–38, find the exact value of the following under the given conditions: c. tan(α + β) 1 3𝝅 1 3𝝅 sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 3 2 3 2
Problem 38
- In Exercises 35–38, find the exact value of the following under the given conditions: d. sin 2α 1 3𝝅 1 3𝝅 sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 3 2 3 2
Problem 38
- In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. cos² 15° - sin² 15°
Problem 39
- In Exercises 43–44, express each product as a sum or difference. sin 6x sin 4x
Problem 43
- In Exercises 43–44, express each product as a sum or difference. sin 7x cos 3x
Problem 44
- In Exercises 45–46, express each sum or difference as a product. If possible, find this product's exact value. sin 2x - sin 4x
Problem 45
- In Exercises 47–54, use the figures to find the exact value of each trigonometric function. θ θ 2 sin ------- cos ------- 2 2
Problem 53
- In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β) 3 5 sin α = ------ , α lies in quadrant I, and sin β = ------- , β lies in quadrant II. 5 13
Problem 57
- In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β) 3 5 sin α = ------ , α lies in quadrant I, and sin β = ------- , β lies in quadrant II. 5 13
Problem 57
- In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β) 8 1 cos α = ------ , α lies in quadrant IV, and sin β = ﹣------- , β lies in quadrant III. 17 2
Problem 61
- In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β) 8 1 cos α = ------ , α lies in quadrant IV, and sin β = ﹣------- , β lies in quadrant III. 17 2
Problem 61
- In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β) 5 𝝅 3 3𝝅 sin α = ------ , -------- < α < 𝝅 , and tan β = ------- , 𝝅 < β < -------- . 6 2 7 2
Problem 64
- In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β) 5 𝝅 3 3𝝅 sin α = ------ , -------- < α < 𝝅 , and tan β = ------- , 𝝅 < β < -------- . 6 2 7 2
Problem 64
- In Exercises 69–74, rewrite each expression as a simplified expression containing one term. sin (α - β) cos β + cos (α - β) sin β
Problem 70