csc(a )=-13 5 11. Find the values of the 6 trig functions of a if Q. III. and a is located in
csc(a )=-13 5 11. Find the values of the 6 trig functions of a if Q. III. and a is located in
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°.
Related questions
Question
![**Problem 11:**
Find the values of the 6 trigonometric functions of \( \alpha \) if \( \csc(\alpha) = -\frac{13}{5} \) and \( \alpha \) is located in Quadrant III.
**Explanation:**
To find the six trigonometric functions, we use the fact that:
- \( \csc(\alpha) = \frac{1}{\sin(\alpha)} \), so \( \sin(\alpha) = -\frac{5}{13} \).
- In Quadrant III, both sine and cosine are negative.
1. **Sine (\(\sin\)):**
\[
\sin(\alpha) = -\frac{5}{13}
\]
2. **Cosine (\(\cos\)):**
Use the Pythagorean identity: \(\sin^2(\alpha) + \cos^2(\alpha) = 1\).
\[
\left(-\frac{5}{13}\right)^2 + \cos^2(\alpha) = 1
\]
\[
\frac{25}{169} + \cos^2(\alpha) = 1
\]
\[
\cos^2(\alpha) = 1 - \frac{25}{169} = \frac{144}{169}
\]
\[
\cos(\alpha) = -\frac{12}{13} \, (\text{since cosine is negative in QIII})
\]
3. **Tangent (\(\tan\)):**
\[
\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} = \frac{-\frac{5}{13}}{-\frac{12}{13}} = \frac{5}{12}
\]
4. **Cosecant (\(\csc\)):**
\[
\csc(\alpha) = -\frac{13}{5}
\]
5. **Secant (\(\sec\)):**
\[
\sec(\alpha) = \frac{1}{\cos(\alpha)} = \frac{1}{-\frac{12}{13}} = -\frac{13}{12}
\]
6. **Cotangent (\(\cot\)):**
\[
\cot(\alpha) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58499046-086a-4660-af23-4374372978d6%2F86acf82a-61d2-4097-828c-638ed50effc8%2Fy05pdnh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 11:**
Find the values of the 6 trigonometric functions of \( \alpha \) if \( \csc(\alpha) = -\frac{13}{5} \) and \( \alpha \) is located in Quadrant III.
**Explanation:**
To find the six trigonometric functions, we use the fact that:
- \( \csc(\alpha) = \frac{1}{\sin(\alpha)} \), so \( \sin(\alpha) = -\frac{5}{13} \).
- In Quadrant III, both sine and cosine are negative.
1. **Sine (\(\sin\)):**
\[
\sin(\alpha) = -\frac{5}{13}
\]
2. **Cosine (\(\cos\)):**
Use the Pythagorean identity: \(\sin^2(\alpha) + \cos^2(\alpha) = 1\).
\[
\left(-\frac{5}{13}\right)^2 + \cos^2(\alpha) = 1
\]
\[
\frac{25}{169} + \cos^2(\alpha) = 1
\]
\[
\cos^2(\alpha) = 1 - \frac{25}{169} = \frac{144}{169}
\]
\[
\cos(\alpha) = -\frac{12}{13} \, (\text{since cosine is negative in QIII})
\]
3. **Tangent (\(\tan\)):**
\[
\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} = \frac{-\frac{5}{13}}{-\frac{12}{13}} = \frac{5}{12}
\]
4. **Cosecant (\(\csc\)):**
\[
\csc(\alpha) = -\frac{13}{5}
\]
5. **Secant (\(\sec\)):**
\[
\sec(\alpha) = \frac{1}{\cos(\alpha)} = \frac{1}{-\frac{12}{13}} = -\frac{13}{12}
\]
6. **Cotangent (\(\cot\)):**
\[
\cot(\alpha) =
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