10. Find the exact value of cos 2x given that cos x =
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°.
Related questions
Question
![**Problem 10:**
Determine the exact value of \( \cos 2x \) given that \( \cos x = \frac{1}{4} \).
**Explanation:**
To find \( \cos 2x \), you can use the double angle identity for cosine:
\[
\cos 2x = 2\cos^2 x - 1
\]
Given \( \cos x = \frac{1}{4} \), substitute this value into the identity:
\[
\cos 2x = 2\left(\frac{1}{4}\right)^2 - 1
\]
Calculate \( \left(\frac{1}{4}\right)^2 \):
\[
\left(\frac{1}{4}\right)^2 = \frac{1}{16}
\]
Substitute back into the equation:
\[
\cos 2x = 2 \cdot \frac{1}{16} - 1
\]
\[
\cos 2x = \frac{2}{16} - 1
\]
\[
\cos 2x = \frac{1}{8} - 1
\]
\[
\cos 2x = \frac{1}{8} - \frac{8}{8}
\]
\[
\cos 2x = \frac{-7}{8}
\]
Therefore, the exact value of \( \cos 2x \) is \( \frac{-7}{8} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5eb9b1d-c2a7-4d17-996f-fe9f500c8923%2F0fcbbe36-ac42-4d7d-a9de-63853d88b116%2Ffg5h3m5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 10:**
Determine the exact value of \( \cos 2x \) given that \( \cos x = \frac{1}{4} \).
**Explanation:**
To find \( \cos 2x \), you can use the double angle identity for cosine:
\[
\cos 2x = 2\cos^2 x - 1
\]
Given \( \cos x = \frac{1}{4} \), substitute this value into the identity:
\[
\cos 2x = 2\left(\frac{1}{4}\right)^2 - 1
\]
Calculate \( \left(\frac{1}{4}\right)^2 \):
\[
\left(\frac{1}{4}\right)^2 = \frac{1}{16}
\]
Substitute back into the equation:
\[
\cos 2x = 2 \cdot \frac{1}{16} - 1
\]
\[
\cos 2x = \frac{2}{16} - 1
\]
\[
\cos 2x = \frac{1}{8} - 1
\]
\[
\cos 2x = \frac{1}{8} - \frac{8}{8}
\]
\[
\cos 2x = \frac{-7}{8}
\]
Therefore, the exact value of \( \cos 2x \) is \( \frac{-7}{8} \).
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