ㅠ If sin x 2 the number A = A cos x, then
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
![If \(\sin \left( x - \frac{\pi}{2} \right) = A \cos x\), then the number \(A =\) [ ].
---
**Explanation:**
This mathematical expression is asking to find the value of \(A\) that satisfies the equation \(\sin \left( x - \frac{\pi}{2} \right) = A \cos x\).
### Steps to Solve:
1. **Use Trigonometric Identity:**
- The identity for the sine of a difference is:
\[
\sin(a - b) = \sin a \cos b - \cos a \sin b
\]
- Apply this identity to \(\sin(x - \frac{\pi}{2})\):
\[
\sin(x) \cos\left(\frac{\pi}{2}\right) - \cos(x) \sin\left(\frac{\pi}{2}\right)
\]
2. **Substitute Known Values:**
- \(\cos\left(\frac{\pi}{2}\right) = 0\)
- \(\sin\left(\frac{\pi}{2}\right) = 1\)
- Therefore:
\[
\sin(x) \cdot 0 - \cos(x) \cdot 1 = -\cos(x)
\]
3. **Equate and Solve:**
- The equation becomes:
\[
-\cos(x) = A \cos(x)
\]
- Divide both sides by \(\cos(x)\) (assuming \(\cos(x) \neq 0\)):
\[
A = -1
\]
Thus, the number \(A\) is \(-1\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1da21a78-616d-4fc2-af3c-7f9c768bf6e4%2Fe02bf8d2-993b-4580-b408-85a73fe9bd47%2Frqqrebe_processed.png&w=3840&q=75)
Transcribed Image Text:If \(\sin \left( x - \frac{\pi}{2} \right) = A \cos x\), then the number \(A =\) [ ].
---
**Explanation:**
This mathematical expression is asking to find the value of \(A\) that satisfies the equation \(\sin \left( x - \frac{\pi}{2} \right) = A \cos x\).
### Steps to Solve:
1. **Use Trigonometric Identity:**
- The identity for the sine of a difference is:
\[
\sin(a - b) = \sin a \cos b - \cos a \sin b
\]
- Apply this identity to \(\sin(x - \frac{\pi}{2})\):
\[
\sin(x) \cos\left(\frac{\pi}{2}\right) - \cos(x) \sin\left(\frac{\pi}{2}\right)
\]
2. **Substitute Known Values:**
- \(\cos\left(\frac{\pi}{2}\right) = 0\)
- \(\sin\left(\frac{\pi}{2}\right) = 1\)
- Therefore:
\[
\sin(x) \cdot 0 - \cos(x) \cdot 1 = -\cos(x)
\]
3. **Equate and Solve:**
- The equation becomes:
\[
-\cos(x) = A \cos(x)
\]
- Divide both sides by \(\cos(x)\) (assuming \(\cos(x) \neq 0\)):
\[
A = -1
\]
Thus, the number \(A\) is \(-1\).
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