# 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}.

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}.