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
![**Trigonometry Problem: Finding Cosine from Sine in Quadrant I**
**Problem Statement:**
Given \(\sin \theta = \frac{\sqrt{48}}{7}\) and angle \(\theta\) is in Quadrant I, what is the exact value of \(\cos \theta\) in simplest form? Simplify all radicals if needed.
**Solution Space:**
**Answer:** [Text box for input]
[Submit Answer button]
---
**Explanation for Students:**
To solve this problem, follow these steps:
1. **Understand the relationship between sine and cosine:**
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
2. **Substitute the given value of \(\sin \theta\) into the equation:**
\[
\left( \frac{\sqrt{48}}{7} \right)^2 + \cos^2 \theta = 1
\]
3. **Simplify the squared sine value:**
\[
\frac{48}{49} + \cos^2 \theta = 1
\]
4. **Solve for \(\cos^2 \theta\):**
\[
\cos^2 \theta = 1 - \frac{48}{49}
\]
\[
\cos^2 \theta = \frac{49}{49} - \frac{48}{49}
\]
\[
\cos^2 \theta = \frac{1}{49}
\]
5. **Take the square root of both sides to find \(\cos \theta\):**
\[
\cos \theta = \pm \frac{1}{7}
\]
Since \(\theta\) is in Quadrant I, where both sine and cosine are positive:
\[
\cos \theta = \frac{1}{7}
\]
**Note:**
Always simplify radicals and express your final answer in the simplest form. Quadrant I consideration is crucial since it affects the sign of the trigonometric functions involved.
**Conclusion:**
The exact value of \(\cos \theta\) is \(\frac{1}{7}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fac6bec-f0b2-4056-963f-cf73547656d5%2F7e18b846-6c2b-44a3-bcda-005cfd688ce0%2Fftv1itb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Trigonometry Problem: Finding Cosine from Sine in Quadrant I**
**Problem Statement:**
Given \(\sin \theta = \frac{\sqrt{48}}{7}\) and angle \(\theta\) is in Quadrant I, what is the exact value of \(\cos \theta\) in simplest form? Simplify all radicals if needed.
**Solution Space:**
**Answer:** [Text box for input]
[Submit Answer button]
---
**Explanation for Students:**
To solve this problem, follow these steps:
1. **Understand the relationship between sine and cosine:**
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
2. **Substitute the given value of \(\sin \theta\) into the equation:**
\[
\left( \frac{\sqrt{48}}{7} \right)^2 + \cos^2 \theta = 1
\]
3. **Simplify the squared sine value:**
\[
\frac{48}{49} + \cos^2 \theta = 1
\]
4. **Solve for \(\cos^2 \theta\):**
\[
\cos^2 \theta = 1 - \frac{48}{49}
\]
\[
\cos^2 \theta = \frac{49}{49} - \frac{48}{49}
\]
\[
\cos^2 \theta = \frac{1}{49}
\]
5. **Take the square root of both sides to find \(\cos \theta\):**
\[
\cos \theta = \pm \frac{1}{7}
\]
Since \(\theta\) is in Quadrant I, where both sine and cosine are positive:
\[
\cos \theta = \frac{1}{7}
\]
**Note:**
Always simplify radicals and express your final answer in the simplest form. Quadrant I consideration is crucial since it affects the sign of the trigonometric functions involved.
**Conclusion:**
The exact value of \(\cos \theta\) is \(\frac{1}{7}\).
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