Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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Find the exact value of sin B(beta)
![Certainly! Here's a transcription suitable for an educational website:
---
**Problem Statement:**
Given the following:
\[
\sin \alpha = \frac{1}{3} \quad \text{and} \quad \frac{\pi}{2} < \alpha < \pi
\]
\[
\cos \beta = -\frac{1}{3} \quad \text{and} \quad \pi < \beta < \frac{3\pi}{2}
\]
---
In this problem, you need to understand the behavior of the trigonometric functions sine and cosine within given intervals on the unit circle.
### Explanation:
1. **Sine Function (\(\sin \alpha\)):**
- \(\sin \alpha = \frac{1}{3}\) suggests that the angle \(\alpha\) corresponds to a position on the unit circle where the y-coordinate is \(\frac{1}{3}\).
- The range \(\frac{\pi}{2} < \alpha < \pi\) indicates that \(\alpha\) is in the second quadrant.
2. **Cosine Function (\(\cos \beta\)):**
- \(\cos \beta = -\frac{1}{3}\) implies that the angle \(\beta\) corresponds to a position where the x-coordinate is \(-\frac{1}{3}\).
- The range \(\pi < \beta < \frac{3\pi}{2}\) indicates that \(\beta\) is in the third quadrant.
These constraints help determine the specific values and signs for trigonometric identities such as sine, cosine, tangent, and their reciprocals within given intervals.](https://content.bartleby.com/qna-images/question/b48d2c1d-e6bb-45e6-b6d7-f9e59d1357b7/c6910b47-1cf1-4cc8-9a80-136e92199507/ttf7pf_processed.jpeg)
Transcribed Image Text:Certainly! Here's a transcription suitable for an educational website:
---
**Problem Statement:**
Given the following:
\[
\sin \alpha = \frac{1}{3} \quad \text{and} \quad \frac{\pi}{2} < \alpha < \pi
\]
\[
\cos \beta = -\frac{1}{3} \quad \text{and} \quad \pi < \beta < \frac{3\pi}{2}
\]
---
In this problem, you need to understand the behavior of the trigonometric functions sine and cosine within given intervals on the unit circle.
### Explanation:
1. **Sine Function (\(\sin \alpha\)):**
- \(\sin \alpha = \frac{1}{3}\) suggests that the angle \(\alpha\) corresponds to a position on the unit circle where the y-coordinate is \(\frac{1}{3}\).
- The range \(\frac{\pi}{2} < \alpha < \pi\) indicates that \(\alpha\) is in the second quadrant.
2. **Cosine Function (\(\cos \beta\)):**
- \(\cos \beta = -\frac{1}{3}\) implies that the angle \(\beta\) corresponds to a position where the x-coordinate is \(-\frac{1}{3}\).
- The range \(\pi < \beta < \frac{3\pi}{2}\) indicates that \(\beta\) is in the third quadrant.
These constraints help determine the specific values and signs for trigonometric identities such as sine, cosine, tangent, and their reciprocals within given intervals.
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