Prove the identity. sin(x - 1) = sin(x) Use the Subtraction Formula for Sine, and then simplify. sin(x - π) = (sin(x))( = (sin(x))( 11 1)-(cos(x) (sin(7)) |)-(cos(x))(0)
Prove the identity. sin(x - 1) = sin(x) Use the Subtraction Formula for Sine, and then simplify. sin(x - π) = (sin(x))( = (sin(x))( 11 1)-(cos(x) (sin(7)) |)-(cos(x))(0)
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°.
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Question
![### Prove the Identity
**Identity:**
\[ \sin(x - \pi) = -\sin(x) \]
**Instructions:**
Use the Subtraction Formula for Sine, and then simplify.
**Step-by-Step Solution:**
1. **Apply the Subtraction Formula for Sine:**
\[ \sin(x - \pi) = \sin(x)\cos(\pi) - \cos(x)\sin(\pi) \]
2. **Substitute the known values:**
\[ \cos(\pi) = -1 \]
\[ \sin(\pi) = 0 \]
Thus, the expression becomes:
\[ \sin(x - \pi) = \sin(x)(-1) - \cos(x)(0) \]
3. **Simplify the expression:**
\[ \sin(x - \pi) = -\sin(x) \]
**Conclusion:**
The original identity \(\sin(x - \pi) = -\sin(x)\) is proved through the Subtraction Formula for Sine.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6841f1bf-ebcf-44b9-bde6-b1a56299f544%2F7be6b784-b455-481e-874b-f6795a228671%2F9yxt8j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Prove the Identity
**Identity:**
\[ \sin(x - \pi) = -\sin(x) \]
**Instructions:**
Use the Subtraction Formula for Sine, and then simplify.
**Step-by-Step Solution:**
1. **Apply the Subtraction Formula for Sine:**
\[ \sin(x - \pi) = \sin(x)\cos(\pi) - \cos(x)\sin(\pi) \]
2. **Substitute the known values:**
\[ \cos(\pi) = -1 \]
\[ \sin(\pi) = 0 \]
Thus, the expression becomes:
\[ \sin(x - \pi) = \sin(x)(-1) - \cos(x)(0) \]
3. **Simplify the expression:**
\[ \sin(x - \pi) = -\sin(x) \]
**Conclusion:**
The original identity \(\sin(x - \pi) = -\sin(x)\) is proved through the Subtraction Formula for Sine.
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