Find the exact value of cot 0. csc 0 21 O in quadrant II 5/21 O A. 21 2/21 O B. 21 V21 OC. V21 O D.
Find the exact value of cot 0. csc 0 21 O in quadrant II 5/21 O A. 21 2/21 O B. 21 V21 OC. V21 O D.
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter1: The Six Trigonometric Functions
Section: Chapter Questions
Problem 11CT
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![### Problem Statement
Find the exact value of \( \cot \theta \).
Given:
\[ \csc \theta = -\frac{5}{2}, \]
where \( \theta \) is in quadrant III.
### Choices:
A. \(-\frac{5\sqrt{21}}{21}\)
B. \(-\frac{2\sqrt{21}}{21}\)
C. \(-\frac{\sqrt{21}}{5}\)
D. \(\frac{\sqrt{21}}{2}\)
## Explanation
To find the value of \( \cot \theta \), we need to recall the trigonometric identities and properties:
1. \(\csc \theta = \frac{1}{\sin \theta}\)
2. \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
Given \(\csc \theta = -\frac{5}{2}\), we can find \(\sin \theta\) as follows:
\[ \sin \theta = -\frac{2}{5} \]
Next, we need to find \(\cos \theta\). Use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substitute \(\sin \theta\):
\[ \left( -\frac{2}{5} \right)^2 + \cos^2 \theta = 1 \]
\[ \frac{4}{25} + \cos^2 \theta = 1 \]
\[ \cos^2 \theta = 1 - \frac{4}{25} \]
\[ \cos^2 \theta = \frac{25}{25} - \frac{4}{25} \]
\[ \cos^2 \theta = \frac{21}{25} \]
\[ \cos \theta = \pm \sqrt{\frac{21}{25}} \]
\[ \cos \theta = \pm \frac{\sqrt{21}}{5} \]
Since \(\theta\) is in quadrant III, both sine and cosine are negative:
\[ \cos \theta = -\frac{\sqrt{21}}{5} \]
Finally, find \(\cot \theta\):
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{-\frac{\sqrt{21}}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad727bdc-baba-4947-a2e3-e934fc451c32%2Fffc7dcd5-a872-4a4e-a761-08ee20f4258c%2Ftbr8dql_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the exact value of \( \cot \theta \).
Given:
\[ \csc \theta = -\frac{5}{2}, \]
where \( \theta \) is in quadrant III.
### Choices:
A. \(-\frac{5\sqrt{21}}{21}\)
B. \(-\frac{2\sqrt{21}}{21}\)
C. \(-\frac{\sqrt{21}}{5}\)
D. \(\frac{\sqrt{21}}{2}\)
## Explanation
To find the value of \( \cot \theta \), we need to recall the trigonometric identities and properties:
1. \(\csc \theta = \frac{1}{\sin \theta}\)
2. \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
Given \(\csc \theta = -\frac{5}{2}\), we can find \(\sin \theta\) as follows:
\[ \sin \theta = -\frac{2}{5} \]
Next, we need to find \(\cos \theta\). Use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substitute \(\sin \theta\):
\[ \left( -\frac{2}{5} \right)^2 + \cos^2 \theta = 1 \]
\[ \frac{4}{25} + \cos^2 \theta = 1 \]
\[ \cos^2 \theta = 1 - \frac{4}{25} \]
\[ \cos^2 \theta = \frac{25}{25} - \frac{4}{25} \]
\[ \cos^2 \theta = \frac{21}{25} \]
\[ \cos \theta = \pm \sqrt{\frac{21}{25}} \]
\[ \cos \theta = \pm \frac{\sqrt{21}}{5} \]
Since \(\theta\) is in quadrant III, both sine and cosine are negative:
\[ \cos \theta = -\frac{\sqrt{21}}{5} \]
Finally, find \(\cot \theta\):
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{-\frac{\sqrt{21}}
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