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 3.3.58a
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
Verified step by step guidance1
Identify the given information: \(\sec \alpha = -3\) and the interval \(\frac{\pi}{2} < \alpha < \pi\).
Recall the relationship between secant and cosine: \(\sec \alpha = \frac{1}{\cos \alpha}\). Use this to find \(\cos \alpha\) by taking the reciprocal of \(\sec \alpha\).
Determine the sign of \(\cos \alpha\) in the given interval. Since \(\frac{\pi}{2} < \alpha < \pi\) corresponds to the second quadrant, where cosine is negative, confirm that \(\cos \alpha\) is negative.
Use the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\) to find \(\sin \alpha\). Substitute the value of \(\cos \alpha\) and solve for \(\sin^2 \alpha\).
Determine the sign of \(\sin \alpha\) in the given interval. Since sine is positive in the second quadrant, take the positive square root to find \(\sin \alpha\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions and Their Relationships
Trigonometric functions like sine, cosine, and secant describe ratios of sides in a right triangle or coordinates on the unit circle. Understanding how secant relates to cosine (sec α = 1/cos α) is essential to find sine values from given secant values.
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6:04
Introduction to Trigonometric Functions
Unit Circle and Angle Quadrants
The unit circle helps determine the sign and value of trig functions based on the angle's quadrant. Knowing that α lies between π/2 and π places it in the second quadrant, where sine is positive and cosine (thus secant) is negative, guiding the correct sign choice.
Recommended video:
06:11Introduction to the Unit Circle
Pythagorean Identity
The Pythagorean identity, sin²α + cos²α = 1, allows calculation of sine when cosine is known. After finding cos α from sec α, use this identity to find sin α, ensuring the correct sign based on the quadrant.
Recommended video:
6:25Pythagorean Identities
Related Practice
Textbook Question
In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. tan x = 2 cos x tan x
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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, 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)
sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.