15 1. If 180° < 0 < 270° and sin 0 = -, find cos . O 2V17 17 17/2 2 17/2 2/17 17 -
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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in image
![### Problem Statement
1. If \( 180^\circ < \theta < 270^\circ \) and \( \sin \theta = -\frac{15}{17} \), find \( \cos \frac{\theta}{2} \).
### Solution Choices
1. \( \frac{2 \sqrt{17}}{17} \)
2. \( \frac{17\sqrt{2}}{2} \)
3. \( -\frac{17\sqrt{2}}{2} \)
4. \( -\frac{2 \sqrt{17}}{17} \)
### Explanation
We are given that the angle \( \theta \) falls within the third quadrant (between 180 degrees and 270 degrees) and that the sine of the angle \( \theta \) is \( -\frac{15}{17} \). Based on this information, we need to determine the value of \( \cos \frac{\theta}{2} \).
1. **Determine the Cosine of \(\theta\):**
Since we know \(\sin \theta\), we can use the Pythagorean identity to find \(\cos \theta\):
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Plugging in the given sine value:
\[
\left(-\frac{15}{17}\right)^2 + \cos^2 \theta = 1
\]
\[
\frac{225}{289} + \cos^2 \theta = 1
\]
\[
\cos^2 \theta = 1 - \frac{225}{289}
\]
\[
\cos^2 \theta = \frac{289 - 225}{289}
\]
\[
\cos \theta = \pm \frac{\sqrt{64}}{17} = \pm \frac{8}{17}
\]
Since \(\theta\) is in the third quadrant, the cosine is negative:
\[
\cos \theta = -\frac{8}{17}
\]
2. **Use the Half-Angle Formula:**
Use the half-angle formula for cosine to find \( \cos \frac{\theta}{2} \):
\[
\cos \frac{\theta}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93030cd0-8340-4952-b4ed-2e3575fbf451%2F7a2bcf26-1a02-40bd-9188-c3b4eeddc117%2F8xm274j_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
1. If \( 180^\circ < \theta < 270^\circ \) and \( \sin \theta = -\frac{15}{17} \), find \( \cos \frac{\theta}{2} \).
### Solution Choices
1. \( \frac{2 \sqrt{17}}{17} \)
2. \( \frac{17\sqrt{2}}{2} \)
3. \( -\frac{17\sqrt{2}}{2} \)
4. \( -\frac{2 \sqrt{17}}{17} \)
### Explanation
We are given that the angle \( \theta \) falls within the third quadrant (between 180 degrees and 270 degrees) and that the sine of the angle \( \theta \) is \( -\frac{15}{17} \). Based on this information, we need to determine the value of \( \cos \frac{\theta}{2} \).
1. **Determine the Cosine of \(\theta\):**
Since we know \(\sin \theta\), we can use the Pythagorean identity to find \(\cos \theta\):
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Plugging in the given sine value:
\[
\left(-\frac{15}{17}\right)^2 + \cos^2 \theta = 1
\]
\[
\frac{225}{289} + \cos^2 \theta = 1
\]
\[
\cos^2 \theta = 1 - \frac{225}{289}
\]
\[
\cos^2 \theta = \frac{289 - 225}{289}
\]
\[
\cos \theta = \pm \frac{\sqrt{64}}{17} = \pm \frac{8}{17}
\]
Since \(\theta\) is in the third quadrant, the cosine is negative:
\[
\cos \theta = -\frac{8}{17}
\]
2. **Use the Half-Angle Formula:**
Use the half-angle formula for cosine to find \( \cos \frac{\theta}{2} \):
\[
\cos \frac{\theta}{
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