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7 sin(2t) + 4 cos(t) = 0 %3D

7 sin(2t) + 4 cos(t) = 0 %3D

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.5: Trigonometric Form For Complex Numbers
Problem 27E
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### 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.
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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