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
Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 3, Problem 3.3.55b
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°
Verified step by step guidance1
Identify the given information: \( \tan \alpha = \frac{4}{3} \) and the angle \( \alpha \) lies in the third quadrant where \( 180^\circ < \alpha < 270^\circ \).
Recall that in the third quadrant, both sine and cosine values are negative, but tangent is positive, which matches the given \( \tan \alpha = \frac{4}{3} \).
Use the identity relating tangent to sine and cosine: \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \). From this, express sine and cosine in terms of a right triangle with opposite side 4 and adjacent side 3.
Calculate the hypotenuse using the Pythagorean theorem: \( r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} \).
Find \( \cos \frac{\alpha}{2} \) using the half-angle formula: \[ \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos \alpha}{2}} \]. Determine the correct sign based on the quadrant of \( \frac{\alpha}{2} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Ratios and Their Definitions
Trigonometric ratios like tangent, sine, and cosine relate the angles of a right triangle to the ratios of its sides. Specifically, tan(α) = opposite/adjacent. Understanding these definitions allows you to find other ratios when one is given.
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Introduction to Trigonometric Functions
Reference Angles and Quadrant Sign Rules
Angles between 180° and 270° lie in the third quadrant, where sine and cosine values are negative, but tangent is positive. Knowing the quadrant helps determine the correct sign of trigonometric values when calculating exact values.
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5:31Reference Angles on the Unit Circle
Half-Angle Formulas
Half-angle formulas express trigonometric functions of half an angle in terms of the original angle, such as cos(α/2) = ±√((1 + cos α)/2). These formulas are essential for finding exact values of trigonometric functions at half the given angle.
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6:36Quadratic Formula
Related Practice
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.
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 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.