How do you evaluate $\left( {\dfrac{{2\pi }}{3}} \right)$ ?

Hint:
We have to find the value of $\cos \dfrac{{2\pi }}{3}$. Firstly we change the angle from $3$ radian to degree. So $\dfrac{{2\pi }}{3}$ will be equal to ${120^{\circ}}$. As the angle ${120^{\circ}}$ is in the second quadrant of the unit circle. and $\cos ine$ is negative and the second quadrant there for value of $\cos {120^{\circ}}$ will be negative. Also ${120^{\circ}}$ is fact opposite of ${60^{\circ}}$ so the value of ${120^{\circ}}$ is equal to the value of $ - \cos {60^{\circ}}$.

Complete step by step solution:
We have to find the value of $\cos \left( {\dfrac{{2\pi }}{3}} \right)?$
Firstly we convert angle in to degree from radian
We know that $\pi {\text{ radian = }}{180^{\circ}}$ .
So $2\pi {\text{ radian = 2}} \times {\text{180 = }}{360^{\circ}}$
Therefore $\dfrac{{2\pi }}{3}{\text{ radian equal to }}{120^{\circ}}$
Now we find$\cos \left( {{{120}^{\circ}}} \right)$.
As it lies in the second quadrant and all the values of the trigonometric function is negative in the second quadrant except $\sin e$ function. Therefore the value of $\cos ine$ is negative in the second quadrant.
Also ${120^{\circ}}$ is just the opposite angle of ${60^{\circ}}$.
So $\cos \left( {{{120}^{\circ}}} \right) = - \cos {60^{\circ}}$
$\cos {60^{\circ}}{\text{ is equal to }}\dfrac{1}{2}$
So value of \[\cos {100^{\circ}}{\text{ = }} - \dfrac{1}{2}\]

Therefore \[\cos \dfrac{{2\pi }}{3}{\text{ = }} - \dfrac{1}{2}\]

Note:
Trigonometry is the branch of mathematics that studies the relationship between side lengths and angles of the triangle. Trigonometry has six trigonometric functions. Which are $\sin ,\cos ,\tan ,\cos ec,\sec and\cot $ . Trigonometric functions are the real functions which relate an angle of right angle triangle to the ratio of two sides of a triangle. Trigonometric functions are also called circular functions. With the help of their trigonometric functions we can drive lots of trigonometric formulas.