Question

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

Answer 1

Hint: 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.,\[ - \dfrac{1}{2}\].