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.35d
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.
Verified step by step guidance1
Identify the given information: \(\sin \alpha = \frac{3}{5}\) with \(0 < \alpha < \frac{\pi}{2}\), and \(\sin \beta = \frac{12}{13}\) with \(\frac{\pi}{2} < \beta < \pi\).
Recall the double-angle identity for sine: \(\sin 2\alpha = 2 \sin \alpha \cos \alpha\).
Since \(\sin \alpha\) is known, find \(\cos \alpha\) using the Pythagorean identity: \(\cos \alpha = \sqrt{1 - \sin^2 \alpha}\), considering the quadrant of \(\alpha\) to determine the sign.
Calculate \(\cos \alpha\) by substituting \(\sin \alpha = \frac{3}{5}\) into the identity: \(\cos \alpha = \sqrt{1 - \left(\frac{3}{5}\right)^2}\).
Substitute \(\sin \alpha\) and \(\cos \alpha\) into the double-angle formula \(\sin 2\alpha = 2 \sin \alpha \cos \alpha\) to express \(\sin 2\alpha\) exactly.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Double-Angle Identity for Sine
The double-angle identity states that sin(2α) = 2 sin(α) cos(α). This formula allows you to find the sine of twice an angle using the sine and cosine of the original angle. Knowing sin(α) and the quadrant of α helps determine cos(α) and thus compute sin(2α) exactly.
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Double Angle Identities
Using the Pythagorean Identity to Find Cosine
Given sin(α), the cosine can be found using cos²(α) = 1 - sin²(α). The sign of cos(α) depends on the quadrant where α lies. Since 0 < α < π/2 (first quadrant), cos(α) is positive, enabling exact calculation of cos(α) from sin(α).
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6:25Pythagorean Identities
Understanding Angle Quadrants and Their Significance
The quadrant of an angle determines the signs of its sine and cosine values. For example, if 0 < α < π/2, both sine and cosine are positive. For β in (π/2, π), sine is positive but cosine is negative. This knowledge is essential for correctly determining trigonometric values and applying identities.
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6:36Quadratic Formula
Related Practice
Textbook Question
In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. tan x = 2 cos x tan x
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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 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.
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