Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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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.
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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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