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.
Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 3, Problem 3.2.59a
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.
Verified step by step guidance1
Identify the given information: \( \tan \alpha = -\frac{3}{4} \) with \( \alpha \) in quadrant II, and \( \cos \beta = \frac{1}{3} \) with \( \beta \) in quadrant I.
Recall the formula for the cosine of a sum:
\[ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \]
Find \( \cos \alpha \) and \( \sin \alpha \) using \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and the quadrant information. Since \( \tan \alpha = -\frac{3}{4} \) and \( \alpha \) is in quadrant II, \( \sin \alpha > 0 \) and \( \cos \alpha < 0 \). Use the Pythagorean identity:
\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]
Express \( \sin \alpha \) and \( \cos \alpha \) in terms of a right triangle with opposite side 3 and adjacent side -4 (due to quadrant II).
Find \( \sin \beta \) using \( \cos \beta = \frac{1}{3} \) and the fact that \( \beta \) is in quadrant I, so \( \sin \beta > 0 \). Use the Pythagorean identity:
\[ \sin^2 \beta + \cos^2 \beta = 1 \]
Substitute the values of \( \cos \alpha \), \( \sin \alpha \), \( \cos \beta \), and \( \sin \beta \) into the cosine sum formula and simplify to find the exact value of \( \cos(\alpha + \beta) \).
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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 Cosine
The sum of angles formula states that cos(α + β) = cos α cos β − sin α sin β. This identity allows us to find the cosine of the sum of two angles using the cosines and sines of the individual angles, which is essential for solving the problem.
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Verifying Identities with Sum and Difference Formulas
Determining Trigonometric Ratios from Given Information
Given tan α and the quadrant of α, we can find sin α and cos α using the Pythagorean identity and sign conventions. Similarly, knowing cos β and the quadrant of β helps determine sin β. Correctly identifying these values is crucial for applying the sum formula.
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6:04Introduction to Trigonometric Functions
Sign of Trigonometric Functions in Different Quadrants
The signs of sine, cosine, and tangent depend on the quadrant of the angle. For example, in quadrant II, sine is positive and cosine is negative; in quadrant I, all are positive. This knowledge ensures correct sign assignment when calculating sin α, cos α, sin β, and cos β.
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Related Practice
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, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅
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 55–58, use the given information to find the exact value of each of the following:
a. sin(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅
4
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Textbook Question
In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.