Blitzer - Trigonometry 3rd Edition - 3장

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Ch. 3 - Trigonometric Identities and Equations

모든 교과서

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Section 3.1: Verifying Trigonometric Identities

문제 11

Verify each identity. csc θ - sin θ = cot θ cos θ

문제 14

Verify each identity. cos θ sec θ/cot θ= tan θ

문제 16

Verify each identity. cos² θ (1 + tan² θ) = 1

문제 22

In Exercises 1–60, verify each identity. cot² t /csc t = csc t - sin t

문제 67

In Exercises 67–74, rewrite each expression in terms of the given function or functions. $\frac{\tan x + \cot x}{\csc x}$ ; $\cos x$

문제 69

In Exercises 67–74, rewrite each expression in terms of the given function or functions. $\frac{\cos x}{1 + \sin x} + \tan x$ ; $\cos x$

문제 71

In Exercises 67–74, rewrite each expression in terms of the given function or functions. $\frac{1}{1 - \cos x} - \frac{\cos x}{1 + \cos x}$ ; $\csc x$

문제 72

In Exercises 67–74, rewrite each expression in terms of the given function or functions. (sec x + csc x) (sin x + cos x) - 2 - cot x; tan x

Section 3.2: Sum and Difference Formulas

문제 1

Use the formula for the cosine of the difference of two angles to solve Exercises 1–12. In Exercises 1–4, find the exact value of each expression. cos(45° - 30°)

문제 3.2.29

In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.

29. sin(5𝝅/12) cos(𝝅/4) - cos(5𝝅/12) sin(𝝅/4)

문제 3.2.63a

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

tan α = 3/4, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅

문제 3.2.59a

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

tan α = ﹣3/4, α lies in quadrant II, and cos β = 1/3, β lies in quadrant I.

문제 3.2.57a

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

문제 3.2.61a

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.

문제 3.2.57b

In Exercises 57–64, find the exact value of the following under the given conditions:

b. sin (α + β)

sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

문제 3.2.61b

In Exercises 57–64, find the exact value of the following under the given conditions:

b. sin (α + β)

cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.

문제 3.2.57c

In Exercises 57–64, find the exact value of the following under the given conditions:

c. tan (α + β)

sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

문제 3.2.61c

In Exercises 57–64, find the exact value of the following under the given conditions:

c. tan (α + β)

cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.

문제 7b

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Write the expression as the cosine of an angle.

$\cos \frac{5 π}{12} \cos \frac{π}{12} + \sin \frac{5 π}{12} \sin \frac{π}{12}$

문제 7c

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Find the exact value of the expression.

$\cos \frac{5 π}{12} \cos \frac{π}{12} + \sin \frac{5 π}{12} \sin \frac{π}{12}$

문제 16

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°

문제 17

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. cos(135° + 30°)

문제 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. tan ( 𝝅/3 + 𝝅/4 )

문제 24

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. tan ( 5𝝅/3 ﹣ 𝝅/4)

문제 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. sin 40° cos 20° + cos 40° sin 20°

문제 64b

In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .

문제 64c

In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .

문제 69

In Exercises 69–74, rewrite each expression as a simplified expression containing one term. cos (α + β) cos β + sin (α + β) sin β

문제 70

In Exercises 69–74, rewrite each expression as a simplified expression containing one term. sin (α - β) cos β + cos (α - β) sin β

Section 3.3: Double-Angle, Power-Reducing, and Half-Angle Formulas

문제 1

In Exercises 1–6, use the figures to find the exact value of each trigonometric function.

sin 2θ