Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions:
e. cos( β/2)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
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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.38c
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
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.
Determine \( \cos \alpha \) using the Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \). Substitute \( \sin \alpha = -\frac{1}{3} \) and solve for \( \cos \alpha \). Since \( \alpha \) is in the third quadrant, \( \cos \alpha \) will be negative.
Determine \( \sin \beta \) using the Pythagorean identity: \( \sin^2 \beta + \cos^2 \beta = 1 \). Substitute \( \cos \beta = -\frac{1}{3} \) and solve for \( \sin \beta \). Since \( \beta \) is in the third quadrant, \( \sin \beta \) will be negative.
Calculate \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \tan \beta = \frac{\sin \beta}{\cos \beta} \) using the values found in previous steps.
Use the tangent addition formula to find \( \tan(\alpha + \beta) \):
\[
\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}
\]
Substitute the values of \( \tan \alpha \) and \( \tan \beta \) 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) using the individual tangents of α and β, which can be derived from their sine and cosine values.
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Sum and Difference of Tangent
Determining Quadrants and Sign of Trigonometric Functions
Knowing the quadrant in which an angle lies is essential to determine the signs of sine, cosine, and tangent. For example, if π < α < 3π/2, α is in the third quadrant where sine and cosine are negative, but tangent is positive. This helps correctly assign signs when calculating tan α and tan β.
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6:04Introduction to Trigonometric Functions
Using Pythagorean Identity to Find Missing Trigonometric Values
Given sin α or cos β, the corresponding cosine α or sine β can be found using the identity sin²θ + cos²θ = 1. This is crucial for computing tan α = sin α / cos α or tan β = sin β / cos β, enabling the use of the sum formula for tangent.
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6:25Pythagorean Identities
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.
699
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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
605
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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
746
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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 = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.
843
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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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