Find all exact solutions for 2 cos z+ 3 sin z on the interval (0, 27). Do not use a calculator for your answers.
Find all exact solutions for 2 cos z+ 3 sin z on the interval (0, 27). Do not use a calculator for your answers.
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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![**Problem Statement:**
Find all exact solutions for \( 2 \cos^3 x + 3 \sin x = 0 \) on the interval \( (0, 2\pi) \). Do not use a calculator for your answers.
**Solution Explanation:**
To solve this trigonometric equation, we'll express it in terms of sine and cosine and solve algebraically within the given interval \( (0, 2\pi) \).
1. **Rearrange the Equation:**
The equation given is:
\[
2 \cos^3 x + 3 \sin x = 0
\]
Isolate the trigonometric terms:
\[
2 \cos^3 x = -3 \sin x
\]
2. **Apply Trigonometric Identities:**
It may be useful to consider transformations such as:
- \(\cos^2 x = 1 - \sin^2 x\)
However, for this equation, further steps can simplify directly.
3. **Factor the Equation:**
Set:
\[
f(x) = 2 \cos^3 x + 3 \sin x
\]
4. **Solve for Roots:**
Substitute potential exact angles where the trigonometric functions have known values, for example, the principal angles.
5. **Verify Solutions:**
Check each solution to ensure it satisfies \( f(x) = 0 \) within the interval \( (0, 2\pi) \).
By following these steps, you can determine the values of \( x \) that satisfy the equation without the use of a calculator.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe6a27354-f783-4962-b110-f2b09bdc3789%2F73d93982-56e7-41b3-b6ff-529471410207%2F5ti9ua2r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find all exact solutions for \( 2 \cos^3 x + 3 \sin x = 0 \) on the interval \( (0, 2\pi) \). Do not use a calculator for your answers.
**Solution Explanation:**
To solve this trigonometric equation, we'll express it in terms of sine and cosine and solve algebraically within the given interval \( (0, 2\pi) \).
1. **Rearrange the Equation:**
The equation given is:
\[
2 \cos^3 x + 3 \sin x = 0
\]
Isolate the trigonometric terms:
\[
2 \cos^3 x = -3 \sin x
\]
2. **Apply Trigonometric Identities:**
It may be useful to consider transformations such as:
- \(\cos^2 x = 1 - \sin^2 x\)
However, for this equation, further steps can simplify directly.
3. **Factor the Equation:**
Set:
\[
f(x) = 2 \cos^3 x + 3 \sin x
\]
4. **Solve for Roots:**
Substitute potential exact angles where the trigonometric functions have known values, for example, the principal angles.
5. **Verify Solutions:**
Check each solution to ensure it satisfies \( f(x) = 0 \) within the interval \( (0, 2\pi) \).
By following these steps, you can determine the values of \( x \) that satisfy the equation without the use of a calculator.
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