Assume sin(0) = Thus, cos(0) = = 49 50 The angle is in quadrant 0 where 2π < 0 < 5. Compute sin(2) 2 - The angle is in quadrant Thus, sin ( 2 ) = .
Assume sin(0) = Thus, cos(0) = = 49 50 The angle is in quadrant 0 where 2π < 0 < 5. Compute sin(2) 2 - The angle is in quadrant Thus, sin ( 2 ) = .
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.5: Product-to-sum And Sum-to-product Formulas
Problem 2E
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![### Example Problem
**Assume \( \sin(\theta) = \frac{49}{50} \) where \( 2\pi < \theta < \frac{5\pi}{2} \). Compute \( \sin\left(\frac{\theta}{2}\right) \).**
**Step-by-Step Solution:**
1. **Determine the quadrant for \( \theta \):**
The angle \( \theta \) is in quadrant
- Quadrant Drop-down menu for students to select the correct quadrant.
2. **Compute \( \cos(\theta) \):**
Since \( \sin(\theta) = \frac{49}{50} \),
\[
\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{49}{50}\right)^2}
\]
3. **Simplify \( \cos(\theta) \):**
\[
\cos(\theta) = \sqrt{1 - \frac{2401}{2500}} = \sqrt{\frac{99}{2500}} = \frac{\sqrt{99}}{50}
\]
Since \( \theta \) is in the fourth quadrant, \( \cos(\theta) \) is positive.
- Answer Box for students to enter \( \cos(\theta) = \frac{\sqrt{99}}{50} \).
4. **Determine the quadrant for \( \frac{\theta}{2} \):**
The angle \( \frac{\theta}{2} \) is in quadrant
- Quadrant Drop-down menu for students to select the correct quadrant.
5. **Compute \( \sin\left(\frac{\theta}{2}\right) \):**
\[
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}
\]
Substituting \( \cos(\theta) \):
\[
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \frac{\sqrt{99}}{50}}{2}}
\]
- Answer Box for students to enter \( \sin\left(\frac{\theta}{2}\right) \).
### Diagrams and Graphs Explanation](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e08cb41-5b2e-4118-8784-a24243109431%2F0486f242-be94-480e-8227-b91a0a69645b%2Fs5098wd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Example Problem
**Assume \( \sin(\theta) = \frac{49}{50} \) where \( 2\pi < \theta < \frac{5\pi}{2} \). Compute \( \sin\left(\frac{\theta}{2}\right) \).**
**Step-by-Step Solution:**
1. **Determine the quadrant for \( \theta \):**
The angle \( \theta \) is in quadrant
- Quadrant Drop-down menu for students to select the correct quadrant.
2. **Compute \( \cos(\theta) \):**
Since \( \sin(\theta) = \frac{49}{50} \),
\[
\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{49}{50}\right)^2}
\]
3. **Simplify \( \cos(\theta) \):**
\[
\cos(\theta) = \sqrt{1 - \frac{2401}{2500}} = \sqrt{\frac{99}{2500}} = \frac{\sqrt{99}}{50}
\]
Since \( \theta \) is in the fourth quadrant, \( \cos(\theta) \) is positive.
- Answer Box for students to enter \( \cos(\theta) = \frac{\sqrt{99}}{50} \).
4. **Determine the quadrant for \( \frac{\theta}{2} \):**
The angle \( \frac{\theta}{2} \) is in quadrant
- Quadrant Drop-down menu for students to select the correct quadrant.
5. **Compute \( \sin\left(\frac{\theta}{2}\right) \):**
\[
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}
\]
Substituting \( \cos(\theta) \):
\[
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \frac{\sqrt{99}}{50}}{2}}
\]
- Answer Box for students to enter \( \sin\left(\frac{\theta}{2}\right) \).
### Diagrams and Graphs Explanation
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