Suppose that sin a = and t

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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## Trigonometric Identities and Equations: Half-angle Identities - Problem Type 2

**Problem Statement:**

Suppose that \( \sin \alpha = -\frac{2}{\sqrt{13}} \) and \( \pi < \alpha < \frac{3\pi}{2} \).

Find the exact values of \( \sin \frac{\alpha}{2} \) and \( \tan \frac{\alpha}{2} \).

**Solution:**

1. **Calculation for \( \sin \frac{\alpha}{2} \):**

   \[
   \sin \frac{\alpha}{2} = \sqrt{\frac{8}{13}}
   \]

   This equation represents the value of the sine of half the angle \(\alpha\).

2. **Calculation for \( \tan \frac{\alpha}{2} \):**

   The tan value isn't provided in the text, typically \( \tan \frac{\alpha}{2} \) can be found using the identity:
   
   \[
   \tan \frac{\alpha}{2} = \frac{1 - \cos \alpha}{\sin \alpha} \quad \text{or another appropriate identity depending on cosine value.}
   \]

Note that these calculations may rely on determining additional trigonometric values not immediately known from \( \sin \alpha \), such as \(\cos \alpha\), typically derived using the Pythagorean identity.

**Graph/Diagram Explanation:**

There is no graph or diagram depicted in this problem. The text involves algebraic manipulation of trigonometric identities to find half-angle values.
Transcribed Image Text:## Trigonometric Identities and Equations: Half-angle Identities - Problem Type 2 **Problem Statement:** Suppose that \( \sin \alpha = -\frac{2}{\sqrt{13}} \) and \( \pi < \alpha < \frac{3\pi}{2} \). Find the exact values of \( \sin \frac{\alpha}{2} \) and \( \tan \frac{\alpha}{2} \). **Solution:** 1. **Calculation for \( \sin \frac{\alpha}{2} \):** \[ \sin \frac{\alpha}{2} = \sqrt{\frac{8}{13}} \] This equation represents the value of the sine of half the angle \(\alpha\). 2. **Calculation for \( \tan \frac{\alpha}{2} \):** The tan value isn't provided in the text, typically \( \tan \frac{\alpha}{2} \) can be found using the identity: \[ \tan \frac{\alpha}{2} = \frac{1 - \cos \alpha}{\sin \alpha} \quad \text{or another appropriate identity depending on cosine value.} \] Note that these calculations may rely on determining additional trigonometric values not immediately known from \( \sin \alpha \), such as \(\cos \alpha\), typically derived using the Pythagorean identity. **Graph/Diagram Explanation:** There is no graph or diagram depicted in this problem. The text involves algebraic manipulation of trigonometric identities to find half-angle values.
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