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...
Related questions
Question
![**Prove the identity.**
\[
\sin(x - \pi) = -\sin(x)
\]
Use the Subtraction Formula for Sine, and then simplify.
\[
\sin(x - \pi) = (\sin(x) \cdot \underline{\hspace{2em}}) - (\cos(x) \cdot \sin(\pi))
\]
\[
= (\sin(x) \cdot \underline{\hspace{2em}}) - (\cos(x) \cdot 0)
\]
\[
= \underline{\hspace{2em}}
\]
**Explanation of Solution Steps:**
To prove the identity \(\sin(x - \pi) = -\sin(x)\), we use the subtraction formula for sine:
1. The subtraction formula is:
\[
\sin(x - y) = \sin(x)\cos(y) - \cos(x)\sin(y)
\]
2. Plugging \(y = \pi\) into the formula:
\[
\sin(x - \pi) = \sin(x)\cos(\pi) - \cos(x)\sin(\pi)
\]
3. We know:
\(\cos(\pi) = -1\) and \(\sin(\pi) = 0\).
4. Substitute these values into the formula:
\[
\sin(x - \pi) = \sin(x)(-1) - \cos(x)(0)
\]
5. Simplify:
\[
= -\sin(x) + 0
= -\sin(x)
\]
Thus, the identity is proven.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90e413d6-0f95-45bc-9120-3de888a1b336%2Fed0ae9c2-3741-40b2-8577-f9ea8c28447c%2Flcqo1us_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Prove the identity.**
\[
\sin(x - \pi) = -\sin(x)
\]
Use the Subtraction Formula for Sine, and then simplify.
\[
\sin(x - \pi) = (\sin(x) \cdot \underline{\hspace{2em}}) - (\cos(x) \cdot \sin(\pi))
\]
\[
= (\sin(x) \cdot \underline{\hspace{2em}}) - (\cos(x) \cdot 0)
\]
\[
= \underline{\hspace{2em}}
\]
**Explanation of Solution Steps:**
To prove the identity \(\sin(x - \pi) = -\sin(x)\), we use the subtraction formula for sine:
1. The subtraction formula is:
\[
\sin(x - y) = \sin(x)\cos(y) - \cos(x)\sin(y)
\]
2. Plugging \(y = \pi\) into the formula:
\[
\sin(x - \pi) = \sin(x)\cos(\pi) - \cos(x)\sin(\pi)
\]
3. We know:
\(\cos(\pi) = -1\) and \(\sin(\pi) = 0\).
4. Substitute these values into the formula:
\[
\sin(x - \pi) = \sin(x)(-1) - \cos(x)(0)
\]
5. Simplify:
\[
= -\sin(x) + 0
= -\sin(x)
\]
Thus, the identity is proven.
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