Find the exact value of each of the remaining trigonometric functions of 0. cos 0 = - 4 e in Quadrant II ..... sin 0 = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
Find the exact value of each of the remaining trigonometric functions of 0. cos 0 = - 4 e in Quadrant II ..... sin 0 = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
9
![**Trigonometric Functions and Exact Values**
**Objective:**
Find the exact value of each of the remaining trigonometric functions of \(\theta\).
Given:
\[
\cos \theta = -\frac{4}{5}
\]
\(\theta\) is located in Quadrant II.
**Question:**
\[
\sin \theta = \underline{\hspace{2cm}}
\]
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
---
### Explanation:
In this task, you are asked to find the sine of an angle \(\theta\) given that the cosine is \(-\frac{4}{5}\) and that the angle is in Quadrant II.
**Key Considerations:**
1. **Cosine and Sine Relationship:**
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
2. **Sign of \(\sin \theta\) in Quadrant II:**
- In Quadrant II, cosine is negative and sine is positive.
3. **Calculating \(\sin \theta\):**
- Start with the Pythagorean identity:
\[
\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(-\frac{4}{5}\right)^2
\]
- Simplify the expression:
\[
\sin^2 \theta = 1 - \left(\frac{16}{25}\right) = \frac{25}{25} - \frac{16}{25} = \frac{9}{25}
\]
- Taking the square root, we get:
\[
\sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5}
\]
- Since \(\theta\) is in Quadrant II, \(\sin \theta\) is positive.
**Conclusion:**
The value of \(\sin \theta\) is \(\frac{3}{5}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd828922-2e6b-41ab-a0d0-1c733c758209%2F6823957a-bdcb-4619-b5b8-d1bd91029939%2F40tq82u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Trigonometric Functions and Exact Values**
**Objective:**
Find the exact value of each of the remaining trigonometric functions of \(\theta\).
Given:
\[
\cos \theta = -\frac{4}{5}
\]
\(\theta\) is located in Quadrant II.
**Question:**
\[
\sin \theta = \underline{\hspace{2cm}}
\]
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
---
### Explanation:
In this task, you are asked to find the sine of an angle \(\theta\) given that the cosine is \(-\frac{4}{5}\) and that the angle is in Quadrant II.
**Key Considerations:**
1. **Cosine and Sine Relationship:**
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
2. **Sign of \(\sin \theta\) in Quadrant II:**
- In Quadrant II, cosine is negative and sine is positive.
3. **Calculating \(\sin \theta\):**
- Start with the Pythagorean identity:
\[
\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(-\frac{4}{5}\right)^2
\]
- Simplify the expression:
\[
\sin^2 \theta = 1 - \left(\frac{16}{25}\right) = \frac{25}{25} - \frac{16}{25} = \frac{9}{25}
\]
- Taking the square root, we get:
\[
\sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5}
\]
- Since \(\theta\) is in Quadrant II, \(\sin \theta\) is positive.
**Conclusion:**
The value of \(\sin \theta\) is \(\frac{3}{5}\).
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