Use the half-angle formula for cosine to compute cos() given cos(0) = 47 50 (Leave your answer in exact form.) where 0 < 0 <
Use the half-angle formula for cosine to compute cos() given cos(0) = 47 50 (Leave your answer in exact form.) where 0 < 0 <
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
![### Half-Angle Formula for Cosine
**Problem Statement:**
Use the half-angle formula for cosine to compute \(\cos\left(\frac{\theta}{2}\right)\) given \(\cos(\theta) = \frac{47}{50}\) where \(0 < \theta < \frac{\pi}{2}\).
**Instructions:**
- Leave your answer in exact form.
**Explanation:**
The half-angle formula for cosine is applied to determine the value of \( \cos\left(\frac{\theta}{2}\right) \). The given problem provides information including the value of \(\cos(\theta)\) and the range within which \(\theta\) lies.
**Formula:**
\[
\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}
\]
Since \(0 < \theta < \frac{\pi}{2}\), the half-angle \(\frac{\theta}{2}\) is also within the first quadrant, meaning \( \cos\left(\frac{\theta}{2}\right) \) is non-negative. Thus the correct form of the formula in this context is:
\[
\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}}
\]
Substitute the given value of \(\cos(\theta)\):
\[
\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{47}{50}}{2}}
\]
**Final Answer:**
\[
\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{47}{50}}{2}}
\]
The answer should be left in this exact form as requested.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e08cb41-5b2e-4118-8784-a24243109431%2F9a14f434-30b6-4a16-ae7a-0df7ead3c4e2%2Fdzuva4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Half-Angle Formula for Cosine
**Problem Statement:**
Use the half-angle formula for cosine to compute \(\cos\left(\frac{\theta}{2}\right)\) given \(\cos(\theta) = \frac{47}{50}\) where \(0 < \theta < \frac{\pi}{2}\).
**Instructions:**
- Leave your answer in exact form.
**Explanation:**
The half-angle formula for cosine is applied to determine the value of \( \cos\left(\frac{\theta}{2}\right) \). The given problem provides information including the value of \(\cos(\theta)\) and the range within which \(\theta\) lies.
**Formula:**
\[
\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}
\]
Since \(0 < \theta < \frac{\pi}{2}\), the half-angle \(\frac{\theta}{2}\) is also within the first quadrant, meaning \( \cos\left(\frac{\theta}{2}\right) \) is non-negative. Thus the correct form of the formula in this context is:
\[
\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}}
\]
Substitute the given value of \(\cos(\theta)\):
\[
\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{47}{50}}{2}}
\]
**Final Answer:**
\[
\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{47}{50}}{2}}
\]
The answer should be left in this exact form as requested.
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