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 3.2.57b
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.
Verified step by step guidance1
Identify the given information: \(\sin \alpha = \frac{3}{5}\) with \(\alpha\) in quadrant I, and \(\sin \beta = \frac{5}{13}\) with \(\beta\) in quadrant II.
Recall the formula for the sine of a sum: \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\).
Find \(\cos \alpha\) using the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\). Since \(\alpha\) is in quadrant I, \(\cos \alpha\) is positive. Calculate \(\cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - \left(\frac{3}{5}\right)^2}\).
Find \(\cos \beta\) similarly using \(\cos \beta = \pm \sqrt{1 - \sin^2 \beta} = \pm \sqrt{1 - \left(\frac{5}{13}\right)^2}\). Since \(\beta\) is in quadrant II, \(\cos \beta\) is negative.
Substitute the values of \(\sin \alpha\), \(\cos \alpha\), \(\sin \beta\), and \(\cos \beta\) into the formula \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\) to find the exact value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sine Addition Formula
The sine addition formula states that sin(α + β) = sin α cos β + cos α sin β. This identity allows us to find the sine of the sum of two angles using the sines and cosines of the individual angles, which is essential for solving the problem.
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6:36
Quadratic Formula
Determining Cosine from Sine and Quadrant
Given sin α and sin β along with their quadrants, we use the Pythagorean identity cos²θ = 1 - sin²θ to find cos α and cos β. The sign of cosine depends on the quadrant: positive in quadrant I and negative in quadrant II, which affects the final calculation.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Quadrant Sign Rules for Trigonometric Functions
The signs of sine and cosine functions vary by quadrant: in quadrant I, both sine and cosine are positive; in quadrant II, sine is positive and cosine is negative. Correctly applying these sign rules is crucial to accurately compute sin(α + β).
Recommended video:
6:04Introduction to Trigonometric Functions
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 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
tan α = ﹣3/4, α lies in quadrant II, and cos β = 1/3, β lies in quadrant I.
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Textbook Question
In Exercises 55–58, use the given information to find the exact value of each of the following:
b. cos(α/2)
tan α = 4/3, 180° < α < 270°
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.
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.