Homework Help is Here – Start Your Trial Now!
Math
Trigonometry
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 (MindTap Course List)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter2: Right Triangle Trigonometry
Section: Chapter Questions
Problem 6GP
See similar textbooks
icon
Related questions
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 \
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 \
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS