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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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 3, Problem 3.2.61a
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.
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 \(\sin \alpha\) using the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\). Since \(\cos \alpha = \frac{8}{17}\), calculate \(\sin \alpha = \pm \sqrt{1 - \left(\frac{8}{17}\right)^2}\). Because \(\alpha\) is in quadrant IV, where sine is negative, choose the negative root.
Determine \(\cos \beta\) using the Pythagorean identity \(\sin^2 \beta + \cos^2 \beta = 1\). Since \(\sin \beta = -\frac{1}{2}\), calculate \(\cos \beta = \pm \sqrt{1 - \left(-\frac{1}{2}\right)^2}\). Because \(\beta\) is in quadrant III, where cosine is negative, choose the negative root.
Use the cosine addition formula: \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\).
Substitute the values of \(\cos \alpha\), \(\cos \beta\), \(\sin \alpha\), and \(\sin \beta\) into the 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 sines and cosines of the individual angles, which is essential for solving the problem.
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Verifying Identities with Sum and Difference Formulas
Determining Sine and Cosine Values from Quadrants
Knowing the quadrant in which an angle lies helps determine the sign of its sine and cosine values. For example, in quadrant IV, cosine is positive and sine is negative; in quadrant III, both sine and cosine are negative. This is crucial for correctly assigning signs to trigonometric values.
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Using Pythagorean Identity to Find Missing Values
The Pythagorean identity, sin²θ + cos²θ = 1, allows calculation of the missing sine or cosine value when one is known. This is important here to find sin α and cos β from the given values, enabling the use of the sum formula accurately.
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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:
a. cos (α + β)
tan α = 3/4, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅
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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
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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.
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Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions:
a. sin(α + β)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
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Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.
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