Find the exact value of cos 0. sin 8= 1- 24 25' πεθε 31 2 ... cos 8= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Find the exact value of cos 0. sin 8= 1- 24 25' πεθε 31 2 ... cos 8= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
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 Statement
Find the exact value of \(\cos \theta\).
Given:
\[
\sin \theta = -\frac{24}{25}, \quad \pi < \theta < \frac{3\pi}{2}
\]
### Answer Box
\[
\cos \theta = \boxed{\phantom{}}
\]
*(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)*
---
### Explanation
The problem requires you to determine the exact value of the cosine of an angle \(\theta\) given the value of its sine and a specific interval \(\pi < \theta < \frac{3\pi}{2}\). The sine and cosine values can be related through the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Given \(\sin \theta = -\frac{24}{25}\), substitute this into the identity and solve for \(\cos \theta\):
\[
\left(-\frac{24}{25}\right)^2 + \cos^2 \theta = 1
\]
Simplify the equation to find \(\cos^2 \theta\), and consider the sign of the cosine based on the given interval. This is crucial as \(\cos \theta\) may be either positive or negative depending on the quadrant in which \(\theta\) lies.
### Detailed Solution
1. Calculate \(\sin^2 \theta\):
\[
\sin^2 \theta = \left(-\frac{24}{25}\right)^2 = \frac{576}{625}
\]
2. Substitute into the Pythagorean identity:
\[
\frac{576}{625} + \cos^2 \theta = 1
\]
3. Solve for \(\cos^2 \theta\):
\[
\cos^2 \theta = 1 - \frac{576}{625} = \frac{625}{625} - \frac{576}{625} = \frac{49}{625}
\]
4. Take the square root of both sides:
\[
\cos \theta = \pm \sqrt{\frac{49}{625}} = \pm \frac{7}{25}
\]
5. Determine the correct sign based on the interval](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe64d88e-699f-422a-bfe6-a9c8b93abe72%2F71bec4c0-4bd1-4d03-8f07-86190e50c5cb%2Fq1ec93_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the exact value of \(\cos \theta\).
Given:
\[
\sin \theta = -\frac{24}{25}, \quad \pi < \theta < \frac{3\pi}{2}
\]
### Answer Box
\[
\cos \theta = \boxed{\phantom{}}
\]
*(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)*
---
### Explanation
The problem requires you to determine the exact value of the cosine of an angle \(\theta\) given the value of its sine and a specific interval \(\pi < \theta < \frac{3\pi}{2}\). The sine and cosine values can be related through the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Given \(\sin \theta = -\frac{24}{25}\), substitute this into the identity and solve for \(\cos \theta\):
\[
\left(-\frac{24}{25}\right)^2 + \cos^2 \theta = 1
\]
Simplify the equation to find \(\cos^2 \theta\), and consider the sign of the cosine based on the given interval. This is crucial as \(\cos \theta\) may be either positive or negative depending on the quadrant in which \(\theta\) lies.
### Detailed Solution
1. Calculate \(\sin^2 \theta\):
\[
\sin^2 \theta = \left(-\frac{24}{25}\right)^2 = \frac{576}{625}
\]
2. Substitute into the Pythagorean identity:
\[
\frac{576}{625} + \cos^2 \theta = 1
\]
3. Solve for \(\cos^2 \theta\):
\[
\cos^2 \theta = 1 - \frac{576}{625} = \frac{625}{625} - \frac{576}{625} = \frac{49}{625}
\]
4. Take the square root of both sides:
\[
\cos \theta = \pm \sqrt{\frac{49}{625}} = \pm \frac{7}{25}
\]
5. Determine the correct sign based on the interval
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