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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

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter5: Trigonometric Functions: Right Triangle Approach
Section5.2: Trigonometry Of Right Triangles
Problem 21E: 21-22Trigonometric Ratios Express x and y in terms of trigonometric ratios of .
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**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 \(\frac{2191}{4050}\).
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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