Problem 5. Use an addition formula to show that v2 cos x- |= cos x+sin x. Graph one period of the function f (x)=cosx + sin x.
Problem 5. Use an addition formula to show that v2 cos x- |= cos x+sin x. Graph one period of the function f (x)=cosx + sin x.
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 5**. Use an addition formula to show that \( \sqrt{2} \cos \left( x - \frac{\pi}{4} \right) = \cos x + \sin x \). Graph one period of the function \( f(x) = \cos x + \sin x \).
---
**Explanation of Concepts:**
1. **Addition Formula for Cosine**:
- The addition formula for cosine states:
\[
\cos(a - b) = \cos a \cos b + \sin a \sin b
\]
- Apply this formula to transform \( \sqrt{2} \cos \left( x - \frac{\pi}{4} \right) \).
2. **Graphing the Function**:
- Graph one period of the function \( f(x) = \cos x + \sin x \).
- Consider the characteristics of the sine and cosine functions to understand the amplitude, period, and behavior over one cycle.
**Visualization Guidance**:
- The graphical representation would typically involve plotting points for the function \( f(x) = \cos x + \sin x \) over the interval \([0, 2\pi]\), indicating peaks, troughs, and intercepts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1b15c0e-b03a-4e06-a4a3-550098e3a000%2Fe6763f05-427d-4921-9518-2106ed5ea076%2Fyzr4da9_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 5**. Use an addition formula to show that \( \sqrt{2} \cos \left( x - \frac{\pi}{4} \right) = \cos x + \sin x \). Graph one period of the function \( f(x) = \cos x + \sin x \).
---
**Explanation of Concepts:**
1. **Addition Formula for Cosine**:
- The addition formula for cosine states:
\[
\cos(a - b) = \cos a \cos b + \sin a \sin b
\]
- Apply this formula to transform \( \sqrt{2} \cos \left( x - \frac{\pi}{4} \right) \).
2. **Graphing the Function**:
- Graph one period of the function \( f(x) = \cos x + \sin x \).
- Consider the characteristics of the sine and cosine functions to understand the amplitude, period, and behavior over one cycle.
**Visualization Guidance**:
- The graphical representation would typically involve plotting points for the function \( f(x) = \cos x + \sin x \) over the interval \([0, 2\pi]\), indicating peaks, troughs, and intercepts.
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