Textbook Question
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.
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 3, Problem 3.2.57c
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.
Verified step by step guidance1
Identify the given values: \(\sin \alpha = \frac{3}{5}\) with \(\alpha\) in quadrant I, and \(\sin \beta = \frac{5}{13}\) with \(\beta\) in quadrant II.
Use the Pythagorean identity to find \(\cos \alpha\) and \(\cos \beta\). Since \(\sin^2 \theta + \cos^2 \theta = 1\), calculate \(\cos \alpha = \sqrt{1 - \sin^2 \alpha}\) and \(\cos \beta = -\sqrt{1 - \sin^2 \beta}\) (negative because \(\beta\) is in quadrant II where cosine is negative).
Calculate \(\tan \alpha\) and \(\tan \beta\) using the definitions \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) with the values found in the previous step.
Apply the tangent addition formula: \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\).
Substitute the values of \(\tan \alpha\) and \(\tan \beta\) into the formula and simplify the expression to find the exact value of \(\tan(\alpha + \beta)\).
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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 like sine, cosine, and tangent relate the angles of a triangle to side lengths. Knowing the quadrant of an angle helps determine the sign (positive or negative) of these ratios, as sine is positive in quadrants I and II, cosine is positive in I and IV, and tangent's sign depends on sine and cosine.
Recommended video:
6:36
Quadratic Formula
Sum of Angles Formula for Tangent
The tangent of a sum of two angles, tan(α + β), can be found using the formula tan(α + β) = (tan α + tan β) / (1 - tan α tan β). This formula allows calculation of the tangent of combined angles from the tangents of individual angles.
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4:47Sum and Difference of Tangent
Finding Missing Trigonometric Ratios Using Pythagorean Identity
Given sin α or sin β, other ratios like cosine and tangent can be found using the Pythagorean identity sin²θ + cos²θ = 1. By solving for cosine and considering the quadrant, one can determine the correct sign and then compute tangent as sin θ / cos θ.
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6:25Pythagorean Identities
Related Practice
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
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.
Textbook Question
In Exercises 1–6, use the figures to find the exact value of each trigonometric function.sin 2α