Textbook Question
In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. sin 22.5°
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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.35c
In Exercises 35–38, find the exact value of the following under the given conditions:
c. tan(α + β)
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 \( \frac{\pi}{2} < \beta < \pi \).
Determine the quadrants for \( \alpha \) and \( \beta \) based on the given interval conditions. Since \( 0 < \alpha < \frac{\pi}{2} \), \( \alpha \) is in the first quadrant where all trigonometric functions are positive. Since \( \frac{\pi}{2} < \beta < \pi \), \( \beta \) is in the second quadrant where sine is positive but cosine is negative.
Find \( \cos \alpha \) using the Pythagorean identity: \( \cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - \left(\frac{3}{5}\right)^2} \). Since \( \alpha \) is in the first quadrant, \( \cos \alpha \) is positive.
Find \( \cos \beta \) similarly: \( \cos \beta = -\sqrt{1 - \sin^2 \beta} = -\sqrt{1 - \left(\frac{12}{13}\right)^2} \). The negative sign is because \( \beta \) is in the second quadrant where cosine is negative.
Use the tangent addition formula: \[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]. Calculate \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \tan \beta = \frac{\sin \beta}{\cos \beta} \), then substitute these values into the formula to express \( \tan(\alpha + \beta) \) exactly.
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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 Tangent
The tangent of the sum of two angles α and β is given by tan(α + β) = (tan α + tan β) / (1 - tan α tan β). This formula allows us to find the exact value of tan(α + β) if we know tan α and tan β, which can be derived from the given sine values.
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Sum and Difference of Tangent
Using Sine to Find Tangent
Given sin α and sin β, we can find cos α and cos β using the Pythagorean identity cos²θ = 1 - sin²θ. Knowing both sine and cosine values allows us to calculate tan θ = sin θ / cos θ. The quadrant information helps determine the correct sign of cosine and tangent.
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Quadrant and Sign Determination
The angles α and β lie in specific quadrants (0 < α < π/2 and π/2 < β < π). The signs of sine, cosine, and tangent depend on the quadrant. For example, in the first quadrant, all are positive, while in the second quadrant, sine is positive but cosine and tangent are negative. Correct sign assignment is crucial for exact values.
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6:36Quadratic Formula
Related Practice
Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
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Textbook Question
In Exercises 45–46, express each sum or difference as a product. If possible, find this product's exact value. sin 2x - sin 4x
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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 43–44, express each product as a sum or difference. sin 6x sin 4x
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Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions:
c. tan(α + β)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2
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