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Given 2\cos 3\theta= 1, find all solutions of the equation.

Question:

Given {eq}2\cos 3\theta= 1 {/eq}, find all solutions of the equation.

Cosine of an angle:

The cosine of an angle can be defined as the fraction of the length of the base of a right angle triangle to the length of the hypotenuse a right angle triangle. The expression for the cosine of an angle is,

{eq}\cos \theta = \dfrac{b}{h} {/eq}

Here, the base of the triangle is {eq}b {/eq} and the hypotenuse of the triangle is {eq}h {/eq}.

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 all solutions for {eq}\theta {/eq} are {eq}\pi /5,\;5\pi /9 {/eq}.


Learn more about this topic:

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How to Find the Period of Cosine Functions

from High School Precalculus: Help and Review

Chapter 24 / Lesson 6
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