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°.
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**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) =
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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