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Given 2\cos 3\theta= 1, find the solutions in the interval [0, 2\pi).

Question:

Given {eq}2\cos 3\theta= 1 {/eq}, find the solutions in the interval {eq}[0, 2\pi). {/eq}

Trigonometric equation:

The trigonometric equation is defined as the type of function that involves one or trigonometric function, mathematical operations, and constant values. The trigonometric equation is used to determine the values of the trigonometric equation.

Answer and Explanation:


The given expression according to the question is,

{eq}2\cos 3\theta = 1 {/eq}


Solve the above expression for the value of \theta .

{eq}\begin{align*} 2\cos 3\theta &= 1\\ \cos 3\theta &= \dfrac{1}{2}\\ 3\theta &= {\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right)\\ 3\theta &= 60^\circ ,\;300^\circ \\ \theta &= 20^\circ ,\;100^\circ \\ &= \pi /5,\;5\pi /9 \end{align*} {/eq}


Thus, the solutions for {eq}\theta {/eq} in the interval {eq}\left[ {0,2\pi } \right] {/eq} are {eq}\pi /5,\;5\pi /9 {/eq}.


Learn more about this topic:

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Solving Trigonometric Equations with Infinite Solutions

from High School Precalculus Textbook

Chapter 22 / Lesson 5
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