4. cos 0 ==, tan 0<0 Find the exact value of sin 0. V33 O A. 7 V33 O B. 4. Ос. -33 O D. -1 4.
4. cos 0 ==, tan 0<0 Find the exact value of sin 0. V33 O A. 7 V33 O B. 4. Ос. -33 O D. -1 4.
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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Question
![### Trigonometry Problem - Finding the Exact Value of sin(θ)
Given:
\[ \cos(\theta) = \frac{4}{7}, \]
\[ \tan(\theta) < 0. \]
Find the exact value of \(\sin(\theta)\).
#### Multiple Choice Options:
A. \(\frac{-\sqrt{33}}{7}\)
B. \(\frac{-\sqrt{33}}{4}\)
C. \(-\sqrt{33}\)
D. \(\frac{-7}{4}\)
---
**Explanation Process:**
1. **Determine Quadrant:**
- Since \(\cos(\theta)\) is positive (\(\cos(\theta) = \frac{4}{7}\)) and \(\tan(\theta)\) is negative, \(\theta\) must be in the fourth quadrant because \(\cos\) is positive and \(\sin\) is negative in that quadrant.
2. **Use Pythagorean Identity:**
- The Pythagorean identity is given by:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
- Substitute \(\cos(\theta) = \frac{4}{7}\):
\[
\sin^2(\theta) + \left(\frac{4}{7}\right)^2 = 1
\]
\[
\sin^2(\theta) + \frac{16}{49} = 1
\]
\[
\sin^2(\theta) = 1 - \frac{16}{49}
\]
\[
\sin^2(\theta) = \frac{49}{49} - \frac{16}{49}
\]
\[
\sin^2(\theta) = \frac{33}{49}
\]
- Taking the square root of both sides and considering the sign of \(\sin(\theta)\) in the fourth quadrant:
\[
\sin(\theta) = -\sqrt{\frac{33}{49}}
\]
\[
\sin(\theta) = -\frac{\sqrt{33}}{7}
\]
Thus, the exact value of \(\sin(\theta)\) is \(\boxed{\frac{-\sqrt{33}}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad727bdc-baba-4947-a2e3-e934fc451c32%2Fc0657149-e4f2-4289-9eb8-e6ce9026fdcc%2Fhflhqxl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometry Problem - Finding the Exact Value of sin(θ)
Given:
\[ \cos(\theta) = \frac{4}{7}, \]
\[ \tan(\theta) < 0. \]
Find the exact value of \(\sin(\theta)\).
#### Multiple Choice Options:
A. \(\frac{-\sqrt{33}}{7}\)
B. \(\frac{-\sqrt{33}}{4}\)
C. \(-\sqrt{33}\)
D. \(\frac{-7}{4}\)
---
**Explanation Process:**
1. **Determine Quadrant:**
- Since \(\cos(\theta)\) is positive (\(\cos(\theta) = \frac{4}{7}\)) and \(\tan(\theta)\) is negative, \(\theta\) must be in the fourth quadrant because \(\cos\) is positive and \(\sin\) is negative in that quadrant.
2. **Use Pythagorean Identity:**
- The Pythagorean identity is given by:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
- Substitute \(\cos(\theta) = \frac{4}{7}\):
\[
\sin^2(\theta) + \left(\frac{4}{7}\right)^2 = 1
\]
\[
\sin^2(\theta) + \frac{16}{49} = 1
\]
\[
\sin^2(\theta) = 1 - \frac{16}{49}
\]
\[
\sin^2(\theta) = \frac{49}{49} - \frac{16}{49}
\]
\[
\sin^2(\theta) = \frac{33}{49}
\]
- Taking the square root of both sides and considering the sign of \(\sin(\theta)\) in the fourth quadrant:
\[
\sin(\theta) = -\sqrt{\frac{33}{49}}
\]
\[
\sin(\theta) = -\frac{\sqrt{33}}{7}
\]
Thus, the exact value of \(\sin(\theta)\) is \(\boxed{\frac{-\sqrt{33}}{
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