Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem Statement
If \( \sin \theta = -\dfrac{3}{5} \) and \( \cot \theta > 0 \), find \( \tan \theta \).
### Multiple Choice Answers
- \( \dfrac{3}{4} \)
- \( -\dfrac{3}{4} \)
- \( \dfrac{4}{3} \)
- \( -\dfrac{4}{3} \)
### Explanation
In this question, you are given the value of \(\sin \theta\) and a condition on \(\cot \theta\).
**Step-by-Step Solution:**
1. We know that:
\( \sin \theta = -\dfrac{3}{5} \)
and \( \cot \theta > 0 \).
2. To find \(\tan \theta\), we need to use the identity involving \(\sin\), \(\cos\), and \(\tan\):
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
3. Substitute \(\sin \theta = -\dfrac{3}{5}\) into the equation:
\[
\left( -\dfrac{3}{5} \right)^2 + \cos^2 \theta = 1
\]
\[
\dfrac{9}{25} + \cos^2 \theta = 1
\]
\[
\cos^2 \theta = 1 - \dfrac{9}{25}
\]
\[
\cos^2 \theta = \dfrac{25}{25} - \dfrac{9}{25}
\]
\[
\cos^2 \theta = \dfrac{16}{25}
\]
\[
\cos \theta = \pm \dfrac{4}{5}
\]
4. Determine the appropriate sign of \(\cos \theta\). Since \(\cot \theta > 0\), both \(\sin \theta\) and \(\cos \theta\) should either be both negative or both positive in the appropriate quadrant:
- Given \(\sin \theta = -\dfrac{3}{5}\), we choose \(\cos \theta = \dfrac{4}{5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1af3b856-d446-4ff7-a63e-89e70c8da553%2F8a43a7e1-e1f4-4a23-bf59-e723d286f337%2Fsr90j6c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
If \( \sin \theta = -\dfrac{3}{5} \) and \( \cot \theta > 0 \), find \( \tan \theta \).
### Multiple Choice Answers
- \( \dfrac{3}{4} \)
- \( -\dfrac{3}{4} \)
- \( \dfrac{4}{3} \)
- \( -\dfrac{4}{3} \)
### Explanation
In this question, you are given the value of \(\sin \theta\) and a condition on \(\cot \theta\).
**Step-by-Step Solution:**
1. We know that:
\( \sin \theta = -\dfrac{3}{5} \)
and \( \cot \theta > 0 \).
2. To find \(\tan \theta\), we need to use the identity involving \(\sin\), \(\cos\), and \(\tan\):
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
3. Substitute \(\sin \theta = -\dfrac{3}{5}\) into the equation:
\[
\left( -\dfrac{3}{5} \right)^2 + \cos^2 \theta = 1
\]
\[
\dfrac{9}{25} + \cos^2 \theta = 1
\]
\[
\cos^2 \theta = 1 - \dfrac{9}{25}
\]
\[
\cos^2 \theta = \dfrac{25}{25} - \dfrac{9}{25}
\]
\[
\cos^2 \theta = \dfrac{16}{25}
\]
\[
\cos \theta = \pm \dfrac{4}{5}
\]
4. Determine the appropriate sign of \(\cos \theta\). Since \(\cot \theta > 0\), both \(\sin \theta\) and \(\cos \theta\) should either be both negative or both positive in the appropriate quadrant:
- Given \(\sin \theta = -\dfrac{3}{5}\), we choose \(\cos \theta = \dfrac{4}{5
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