Textbook Question
In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. cos² 15° - sin² 15°
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 3, Problem 3.RE.38a
In Exercises 35–38, find the exact value of the following under the given conditions:
a. sin(α + β)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
Verified step by step guidance1
Identify the given information: \( \sin \alpha = -\frac{1}{3} \) with \( \pi < \alpha < \frac{3\pi}{2} \), and \( \cos \beta = -\frac{1}{3} \) with \( \pi < \beta < \frac{3\pi}{2} \). Both angles are in the third quadrant.
Recall the sine addition formula: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \). We need to find \( \cos \alpha \) and \( \sin \beta \) to use this formula.
Use the Pythagorean identity to find \( \cos \alpha \): \( \cos \alpha = -\sqrt{1 - \sin^2 \alpha} = -\sqrt{1 - \left(-\frac{1}{3}\right)^2} \). The negative sign is because \( \alpha \) is in the third quadrant where cosine is negative.
Similarly, find \( \sin \beta \) using the identity \( \sin \beta = -\sqrt{1 - \cos^2 \beta} = -\sqrt{1 - \left(-\frac{1}{3}\right)^2} \), since \( \beta \) is also in the third quadrant where sine is negative.
Substitute all known values into the sine addition formula: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \), and simplify the expression to find the exact value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sum of Angles Formula for Sine
The sum of angles formula states that sin(α + β) = sin α cos β + cos α sin β. This identity allows us to find the sine of a sum of two angles using the sines and cosines of the individual angles, which is essential for solving the problem.
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Verifying Identities with Sum and Difference Formulas
Determining Cosine and Sine from Given Values and Quadrants
Given sin α and cos β along with their quadrant information, we use the Pythagorean identity (sin²θ + cos²θ = 1) to find the missing cosine or sine values. The quadrant determines the sign (positive or negative) of these values, which is crucial for accuracy.
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Understanding Angle Measures and Quadrants
The problem specifies angle ranges (π < α < 3π/2 and π < β < 3π/2), indicating both angles lie in the third quadrant. In this quadrant, sine and cosine values are negative, which affects the sign of the trigonometric functions and must be considered when calculating exact values.
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Related Practice
Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions:
e. cos(β/2)
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.
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Textbook Question
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.
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Textbook Question
In Exercises 55–58, use the given information to find the exact value of each of the following:
a. sin(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅
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Textbook Question
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.
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Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.
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