Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Rewrite trigonometric identity
#### Instructions
**Rewrite**
\[ \sin \left( x + \frac{\pi}{6} \right) \]
in terms of \(\sin x\) and \(\cos x\).
**Note**: Enclose arguments of functions in parentheses. For example, \(\sin (2x)\).
\[ \sin \left( x + \frac{\pi}{6} \right) = \]
(Insert your solution in the box provided)
---
#### Explanation
To rewrite the given sine function using \(\sin x\) and \(\cos x\), you will need to use the angle addition formula for sine:
\[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \]
By substituting \(a = x\) and \(b = \frac{\pi}{6}\) and knowing the values:
\[ \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \]
\[ \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \]
You get:
\[ \sin \left( x + \frac{\pi}{6} \right) = \sin(x)\cos\left( \frac{\pi}{6} \right) + \cos(x)\sin\left( \frac{\pi}{6} \right) \]
\[ = \sin(x) \cdot \frac{\sqrt{3}}{2} + \cos(x) \cdot \frac{1}{2} \]
\[ = \frac{\sqrt{3}}{2} \sin(x) + \frac{1}{2} \cos(x) \]
Therefore,
\[ \sin \left( x + \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \sin(x) + \frac{1}{2} \cos(x) \]
Make sure you enter the correct expression in the provided input box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F704f324a-66c2-47cf-962c-2947999b0de6%2F39b4f626-1f5a-4321-84c3-050d0d5adf6b%2Futldmqi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Rewrite trigonometric identity
#### Instructions
**Rewrite**
\[ \sin \left( x + \frac{\pi}{6} \right) \]
in terms of \(\sin x\) and \(\cos x\).
**Note**: Enclose arguments of functions in parentheses. For example, \(\sin (2x)\).
\[ \sin \left( x + \frac{\pi}{6} \right) = \]
(Insert your solution in the box provided)
---
#### Explanation
To rewrite the given sine function using \(\sin x\) and \(\cos x\), you will need to use the angle addition formula for sine:
\[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \]
By substituting \(a = x\) and \(b = \frac{\pi}{6}\) and knowing the values:
\[ \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \]
\[ \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \]
You get:
\[ \sin \left( x + \frac{\pi}{6} \right) = \sin(x)\cos\left( \frac{\pi}{6} \right) + \cos(x)\sin\left( \frac{\pi}{6} \right) \]
\[ = \sin(x) \cdot \frac{\sqrt{3}}{2} + \cos(x) \cdot \frac{1}{2} \]
\[ = \frac{\sqrt{3}}{2} \sin(x) + \frac{1}{2} \cos(x) \]
Therefore,
\[ \sin \left( x + \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \sin(x) + \frac{1}{2} \cos(x) \]
Make sure you enter the correct expression in the provided input box.
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