Use the double-angle formula for cosine to compute cos(20) given cos( os(0) = where 0 <0<. 79 90
Use the double-angle formula for cosine to compute cos(20) given cos( os(0) = where 0 <0<. 79 90
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Question
![**Application of the Double-Angle Formula for Cosine**
To compute \(\cos(2\theta)\) when given \(\cos(\theta) = \frac{79}{90}\) and the range \(0 < \theta < \frac{\pi}{2}\), we use the double-angle formula for cosine.
The double-angle formula for cosine states:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
Given \(\cos(\theta) = \frac{79}{90}\), we substitute this value into the formula:
1. First, find \(\cos^2(\theta)\):
\[ \cos^2(\theta) = \left(\frac{79}{90}\right)^2 \]
2. Compute the value:
\[ \cos^2(\theta) = \frac{6241}{8100} \]
3. Now, apply the double-angle formula:
\[ \cos(2\theta) = 2\left(\frac{6241}{8100}\right) - 1 \]
\[ \cos(2\theta) = \frac{12482}{8100} - 1 \]
\[ \cos(2\theta) = \frac{12482}{8100} - \frac{8100}{8100} \]
\[ \cos(2\theta) = \frac{12482 - 8100}{8100} \]
\[ \cos(2\theta) = \frac{4382}{8100} \]
Simplify the fraction, if possible:
\[ \cos(2\theta) = \frac{2191}{4050} \]
Thus, the value of \(\cos(2\theta)\) is \(\frac{2191}{4050}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e08cb41-5b2e-4118-8784-a24243109431%2Fa442ac50-389a-449f-9321-b98f21cd8fda%2Fzlnicb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Application of the Double-Angle Formula for Cosine**
To compute \(\cos(2\theta)\) when given \(\cos(\theta) = \frac{79}{90}\) and the range \(0 < \theta < \frac{\pi}{2}\), we use the double-angle formula for cosine.
The double-angle formula for cosine states:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
Given \(\cos(\theta) = \frac{79}{90}\), we substitute this value into the formula:
1. First, find \(\cos^2(\theta)\):
\[ \cos^2(\theta) = \left(\frac{79}{90}\right)^2 \]
2. Compute the value:
\[ \cos^2(\theta) = \frac{6241}{8100} \]
3. Now, apply the double-angle formula:
\[ \cos(2\theta) = 2\left(\frac{6241}{8100}\right) - 1 \]
\[ \cos(2\theta) = \frac{12482}{8100} - 1 \]
\[ \cos(2\theta) = \frac{12482}{8100} - \frac{8100}{8100} \]
\[ \cos(2\theta) = \frac{12482 - 8100}{8100} \]
\[ \cos(2\theta) = \frac{4382}{8100} \]
Simplify the fraction, if possible:
\[ \cos(2\theta) = \frac{2191}{4050} \]
Thus, the value of \(\cos(2\theta)\) is \(\frac{2191}{4050}\).
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