Textbook Question
In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. cos² 15° - sin² 15°
Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 3, Problem 3.2.57a
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.
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 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 \(\cos \beta = \pm \sqrt{1 - \sin^2 \beta}\).
Determine the correct signs for \(\cos \alpha\) and \(\cos \beta\) based on the quadrant information: in quadrant I, cosine is positive; in quadrant II, cosine is negative.
Use the cosine addition formula: \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\).
Substitute the values of \(\sin \alpha\), \(\cos \alpha\), \(\sin \beta\), and \(\cos \beta\) 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 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 value.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Quadrant Sign Rules for Trigonometric Functions
The sign of sine and cosine functions depends on the quadrant of the angle. In quadrant I, both sine and cosine are positive; in quadrant II, sine is positive but cosine is negative. Correctly applying these sign rules is crucial for accurate calculation.
Recommended video:
6:04Introduction to Trigonometric Functions
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, α lies in quadrant II, and cos β = 1/3, β lies in quadrant I.
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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)
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:
a. sin(α/2)
tan α = 4/3, 180° < α < 270°
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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)
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
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