Question

How do you find the exact value of $\cos \left( {\dfrac{{2\pi }}{3}} \right)$?

Answer 1

In this question we have to find the value of cos value of the angle given, this can be done by using trigonometric double angle identity i.e.,$\cos 2A = 2{\cos ^2}A - 1$, and we should know that the value of $\cos \left( {\dfrac{\pi }{3}} \right) = \dfrac{1}{2}$, and substituting the values in the identities we will get the required value.

Complete step-by-step solution:
Given trigonometric expression is $\cos \left( {\dfrac{{2\pi }}{3}} \right)$,
Now transforming the expression in form of $\cos 2A$, we get,
$\Rightarrow \cos 2\left( {\dfrac{\pi }{3}} \right)$,
We know that the trigonometric identity for the double angle for cos which is given by,
$\cos 2A = 2{\cos ^2}A - 1$,
Now comparing two expressions we get, here$A = \dfrac{\pi }{3}$,b
By substituting the value in the trigonometric identity, $\cos 2A = 2{\cos ^2}A - 1$, we get,
$\Rightarrow \cos 2\left( {\dfrac{\pi }{3}} \right) = 2{\cos ^2}\dfrac{\pi }{3} - 1$,
We know that$\cos \left( {\dfrac{\pi }{3}} \right) = \dfrac{1}{2}$, now substituting the value in the identity we get,
$\Rightarrow \cos 2\left( {\dfrac{\pi }{3}} \right) = 2{\left( {\dfrac{1}{2}} \right)^2} - 1$,
So, now simplifying we get,
$\Rightarrow \cos 2\left( {\dfrac{\pi }{3}} \right) = 2\left( {\dfrac{1}{4}} \right) - 1$,
Now removing the brackets by doing multiplication we get,
$\Rightarrow \cos 2\left( {\dfrac{\pi }{3}} \right) = \dfrac{1}{2} - 1$,
Now taking the L.C.M on the right hand side we get,
$\Rightarrow \cos 2\left( {\dfrac{\pi }{3}} \right) = \dfrac{{1 - 2}}{2}$,
By further simplification we get,
$\Rightarrow \cos 2\left( {\dfrac{\pi }{3}} \right) = \dfrac{{ - 1}}{2}$,
So, the exact value for the cos is $\dfrac{{ - 1}}{2}$.

$\therefore$The exact value for the given cos angle i.e., $\cos \left( {\dfrac{{2\pi }}{3}} \right)$ will be equal to $\dfrac{{ - 1}}{2}$.

Note: This question can be solved by another method by using the identity $\cos \left( {\pi - \theta } \right) = - \cos \theta$, so here $\dfrac{{2\pi }}{3}$ can be written as,$\pi - \dfrac{\pi }{3}$, i.e,
$\Rightarrow \dfrac{{2\pi }}{3} = \pi - \dfrac{\pi }{3}$,
Now applying cos on both sides we get,
$\Rightarrow \cos \dfrac{{2\pi }}{3} = \cos \left( {\pi - \dfrac{\pi }{3}} \right)$,
We know that $\cos \left( {\pi - \theta } \right) = - \cos \theta$, now applying the identity we get,
$\Rightarrow \cos \dfrac{{2\pi }}{3} = - \cos \left( {\dfrac{\pi }{3}} \right)$,
And we know that $\cos \left( {\dfrac{\pi }{3}} \right) = \dfrac{1}{2}$, now substituting the value in the expression we get,
$\Rightarrow \cos \dfrac{{2\pi }}{3} = - \dfrac{1}{2}$,
So from the two methods we got the same value for $\cos \dfrac{{2\pi }}{3}$ i.e.,$- \dfrac{1}{2}$.