Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.4: Multiple-angle Formulas
Problem 41E
Related questions
Question
![### Problem Statement
**1. Solve on \([0, 2\pi)\)**
\[ 7 \sin(2t) + 4 \cos(t) = 0 \]
### Explanation
This problem asks to find the values of \(t\) in the interval from \(0\) to \(2\pi\) (including \(0\) but excluding \(2\pi\)) that satisfy the equation. The equation involves a trigonometric identity and requires manipulation to solve for \(t\).
### Steps to Solve
1. **Double Angle Identity for Sine**: Use the identity \(\sin(2t) = 2\sin(t)\cos(t)\) to rewrite the equation.
2. **Substitute and Simplify**: Substitute \(\sin(2t)\) in the equation and simplify to make the equation easier to solve.
3. **Solve for \(t\)**: Find solutions for \(t\) within the given interval by finding values that satisfy the simplified equation.
This problem involves knowledge of trigonometric identities and equation-solving techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6735299-61de-45a1-8389-cd026716d460%2F4de6dd25-13fb-4053-8761-2b6b5651ca23%2F8u18hza_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**1. Solve on \([0, 2\pi)\)**
\[ 7 \sin(2t) + 4 \cos(t) = 0 \]
### Explanation
This problem asks to find the values of \(t\) in the interval from \(0\) to \(2\pi\) (including \(0\) but excluding \(2\pi\)) that satisfy the equation. The equation involves a trigonometric identity and requires manipulation to solve for \(t\).
### Steps to Solve
1. **Double Angle Identity for Sine**: Use the identity \(\sin(2t) = 2\sin(t)\cos(t)\) to rewrite the equation.
2. **Substitute and Simplify**: Substitute \(\sin(2t)\) in the equation and simplify to make the equation easier to solve.
3. **Solve for \(t\)**: Find solutions for \(t\) within the given interval by finding values that satisfy the simplified equation.
This problem involves knowledge of trigonometric identities and equation-solving techniques.
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