Textbook Question
In Exercises 55–58, use the given information to find the exact value of each of the following:
b. cos(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅
Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 3, Problem 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.
Verified step by step guidance1
Identify the given information: \(\cos \alpha = \frac{8}{17}\) with \(\alpha\) in quadrant IV, and \(\sin \beta = -\frac{1}{2}\) with \(\beta\) in quadrant III.
Determine the signs and values of \(\sin \alpha\) and \(\cos \beta\) using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), considering the quadrant of each angle.
Calculate \(\sin \alpha\) by using \(\sin \alpha = -\sqrt{1 - \cos^2 \alpha}\) since \(\alpha\) is in quadrant IV where sine is negative.
Calculate \(\cos \beta\) by using \(\cos \beta = -\sqrt{1 - \sin^2 \beta}\) since \(\beta\) is in quadrant III where cosine is negative.
Use the angle addition formula for tangent: \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\), where \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\) and \(\tan \beta = \frac{\sin \beta}{\cos \beta}\). Substitute the values found to express \(\tan(\alpha + \beta)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Ratios and Quadrants
Trigonometric ratios (sine, cosine, tangent) relate the angles of a triangle to the ratios of its sides. The sign of these ratios depends on the quadrant in which the angle lies. For example, in quadrant IV, cosine is positive and sine is negative, while in quadrant III, both sine and cosine are negative.
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Quadratic Formula
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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4:47Sum and Difference of Tangent
Finding Missing Trigonometric Values Using Pythagorean Identity
Given one trigonometric ratio and the quadrant, the other ratios can be found using the Pythagorean identity sin²θ + cos²θ = 1. For example, if cos α is known, sin α can be found by sin α = ±√(1 - cos²α), with the sign determined by the quadrant. This step is essential to compute tan α and tan β.
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6:25Pythagorean Identities
Related Practice
Textbook Question
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.
1
views
Textbook Question
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.
Textbook Question
In Exercises 55–58, use the given information to find the exact value of each of the following:
c. tan(α/2)
tan α = 4/3, 180° < α < 270°
5
views
Textbook Question
In Exercises 55–58, use the given information to find the exact value of each of the following:
c. tan(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅
3
views
Textbook Question
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.