Question

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

Answer 1

Hint: Here we will use the All STC rule and the trigonometric values to find the value of the given function. First of all will convert the given angle in the equivalent angle form and then place the values.

Complete step-by-step solution:
Take the given expression: $\cos \left( {\dfrac{{2\pi }}{3}} \right)$
Convert the given angle in the equivalent angle form -
$\cos \left( {\dfrac{{2\pi }}{3}} \right) = \cos \left( {\pi - \dfrac{{2\pi }}{3}} \right)$
Now, the given angle lies in the second quadrant and by all STC rules cosine is negative in the second quadrant.
$\cos \left( {\dfrac{{2\pi }}{3}} \right) = - \cos \left( {\dfrac{\pi }{3}} \right)$
Refer the trigonometric table for the reference value of the above angle.
$\cos \left( {\dfrac{{2\pi }}{3}} \right) = - \dfrac{1}{2}$
This is the required solution.

Thus the required solution $\cos \left( {\dfrac{{2\pi }}{3}} \right) = - \dfrac{1}{2}$.

Note: Remember the All STC rule, it is also known as ASTC rule in geometry. It states that all the trigonometric ratios in the first quadrant ($0^\circ \;{\text{to 90}}^\circ $ ) are positive, sine and cosec are positive in the second quadrant ($90^\circ {\text{ to 180}}^\circ $ ), tan and cot are positive in the third quadrant ($180^\circ \;{\text{to 270}}^\circ $ ) and sin and cosec are positive in the fourth quadrant ($270^\circ {\text{ to 360}}^\circ $ ). Remember the trigonometric table for the reference values for different angles for sine, cosine and tangent functions for direct substitution.