Suppose that sin0 COS tan 2 Find the exact values of cos and tan 2 0 2 = 0 = = - 0 5 13 and 0°<0<90°. 0 2 8 Undefined X √67 S
Suppose that sin0 COS tan 2 Find the exact values of cos and tan 2 0 2 = 0 = = - 0 5 13 and 0°<0<90°. 0 2 8 Undefined X √67 S
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
![### Trigonometric Identities and Half-Angle Formulas
**Problem Statement:**
Suppose that \(\sin \theta = \frac{5}{13}\) and \(0^\circ < \theta < 90^\circ\).
Find the exact values of \(\cos \frac{\theta}{2}\) and \(\tan \frac{\theta}{2}\).
**Solution:**
1. **Given:**
\(\sin \theta = \frac{5}{13}\)
2. **Objective:**
Find:
\[
\cos \frac{\theta}{2} \quad \text{and} \quad \tan \frac{\theta}{2}
\]
3. **Using the Half-Angle Formulas:**
For \(\cos \frac{\theta}{2}\):
\[
\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}
\]
For \(\tan \frac{\theta}{2}\):
\[
\tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}
\]
4. **Find \(\cos \theta\):**
\[
\cos^2 \theta + \sin^2 \theta = 1
\]
\[
\cos^2 \theta + \left(\frac{5}{13}\right)^2 = 1
\]
\[
\cos^2 \theta + \frac{25}{169} = 1
\]
\[
\cos^2 \theta = 1 - \frac{25}{169}
\]
\[
\cos^2 \theta = \frac{144}{169}
\]
\[
\cos \theta = \pm \frac{12}{13}
\]
Since \(0^\circ < \theta < 90^\circ\), \(\cos \theta\) is positive:
\[
\cos \theta = \frac{12}{13}
\]
5. **Calculate \(\cos \frac{\theta}{2}\):**
\[
\cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a948ce4-bfec-4c4b-a693-d4721e6497c9%2F3b97c105-5350-4536-bfe8-0d3a8ee40e45%2Fc3c1o5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometric Identities and Half-Angle Formulas
**Problem Statement:**
Suppose that \(\sin \theta = \frac{5}{13}\) and \(0^\circ < \theta < 90^\circ\).
Find the exact values of \(\cos \frac{\theta}{2}\) and \(\tan \frac{\theta}{2}\).
**Solution:**
1. **Given:**
\(\sin \theta = \frac{5}{13}\)
2. **Objective:**
Find:
\[
\cos \frac{\theta}{2} \quad \text{and} \quad \tan \frac{\theta}{2}
\]
3. **Using the Half-Angle Formulas:**
For \(\cos \frac{\theta}{2}\):
\[
\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}
\]
For \(\tan \frac{\theta}{2}\):
\[
\tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}
\]
4. **Find \(\cos \theta\):**
\[
\cos^2 \theta + \sin^2 \theta = 1
\]
\[
\cos^2 \theta + \left(\frac{5}{13}\right)^2 = 1
\]
\[
\cos^2 \theta + \frac{25}{169} = 1
\]
\[
\cos^2 \theta = 1 - \frac{25}{169}
\]
\[
\cos^2 \theta = \frac{144}{169}
\]
\[
\cos \theta = \pm \frac{12}{13}
\]
Since \(0^\circ < \theta < 90^\circ\), \(\cos \theta\) is positive:
\[
\cos \theta = \frac{12}{13}
\]
5. **Calculate \(\cos \frac{\theta}{2}\):**
\[
\cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \
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