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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![### Problem Description
**Find all solutions on the interval \( [0^\circ, 360^\circ) \). Use exact values.**
\[3 \cos x - 3 \sin x \cos x = 0\]
Separate multiple solutions with a comma. If there are no solutions please enter, "no solution" exactly.
\[ \_\_\_\_\_\_ \]
### Steps and Tips
To solve the equation \( 3 \cos x - 3 \sin x \cos x = 0 \):
1. **Simplify the Equation:**
- Factor out the common term:
\[
3 \cos x (1 - \sin x) = 0
\]
2. **Solve for Each Factor:**
- The equation is satisfied when \( 3 \cos x = 0 \) or \( 1 - \sin x = 0 \).
3. **For \( \cos x = 0 \):**
- Find values of \( x \) in the interval \( [0^\circ, 360^\circ) \).
- \[
\cos x = 0 \implies x = 90^\circ, 270^\circ
\]
4. **For \( 1 - \sin x = 0 \):**
- Simplify to find \( \sin x = 1 \).
- Find values of \( x \) in the interval \( [0^\circ, 360^\circ) \).
- \[
\sin x = 1 \implies x = 90^\circ
\]
5. **Combine Solutions:**
- List all unique solutions found:
\[
x = 90^\circ, 270^\circ
\]
### Solution
Enter the solutions in a comma-separated list:
\[ 90^\circ, 270^\circ \]
If the equation had no solutions, you would enter "no solution" exactly. However, in this case, we have the solutions!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9546f28b-d85a-4d28-aa6c-948f4880bd46%2Fd9dc0f0a-3ec9-4b8e-b208-bb31bdaa5ba7%2Fwcqmvyy_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Description
**Find all solutions on the interval \( [0^\circ, 360^\circ) \). Use exact values.**
\[3 \cos x - 3 \sin x \cos x = 0\]
Separate multiple solutions with a comma. If there are no solutions please enter, "no solution" exactly.
\[ \_\_\_\_\_\_ \]
### Steps and Tips
To solve the equation \( 3 \cos x - 3 \sin x \cos x = 0 \):
1. **Simplify the Equation:**
- Factor out the common term:
\[
3 \cos x (1 - \sin x) = 0
\]
2. **Solve for Each Factor:**
- The equation is satisfied when \( 3 \cos x = 0 \) or \( 1 - \sin x = 0 \).
3. **For \( \cos x = 0 \):**
- Find values of \( x \) in the interval \( [0^\circ, 360^\circ) \).
- \[
\cos x = 0 \implies x = 90^\circ, 270^\circ
\]
4. **For \( 1 - \sin x = 0 \):**
- Simplify to find \( \sin x = 1 \).
- Find values of \( x \) in the interval \( [0^\circ, 360^\circ) \).
- \[
\sin x = 1 \implies x = 90^\circ
\]
5. **Combine Solutions:**
- List all unique solutions found:
\[
x = 90^\circ, 270^\circ
\]
### Solution
Enter the solutions in a comma-separated list:
\[ 90^\circ, 270^\circ \]
If the equation had no solutions, you would enter "no solution" exactly. However, in this case, we have the solutions!
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