Question

How do you find the exact value of $\tan \left( {\dfrac{{5\pi }}{3}} \right)$ ?

Answer 1

Hint:Tangent repeats itself at the multiples of $\pi $. Represent the angle using $2\pi $ to get a negative angle that represents the clockwise movement. Use $\tan \left( { - x} \right) = - \tan x$ and then the definition of the tangent ratio, i.e. $\tan x = \dfrac{{\sin x}}{{\cos x}}$ . This will give you the required value of $\tan \left( {\dfrac{{5\pi }}{3}} \right)$.

Complete step by step answer:
Here in this problem, we are given an angle of $\dfrac{{5\pi }}{3}$ radians and we need to find the value for tangent function for this angle, i.e. $\tan \left( {\dfrac{{5\pi }}{3}} \right)$ . We can use some trigonometric properties to solve this problem.
Before starting with the problem we need to understand a few concepts related to this situation. Tan function (or tangent function) in a triangle is the ratio of the opposite side to that of the adjacent side. The tangent function is one of the three main primary trigonometric functions. In a right-triangle, tan is defined as the ratio of the length of the perpendicular side to that of the adjacent side i.e. the base.
As we know, $\pi {\text{ }}radians = 180^\circ $, therefore the given angle can be represented in degree as:
$ \Rightarrow \pi {\text{ }}radians = 180^\circ \Rightarrow \dfrac{{5\pi }}{3} = 180^\circ \times \dfrac{5}{3} = 60^\circ \times 5 = 300^\circ $
So we need to evaluate the expression $\tan 300^\circ $.
This can also be represented as:
$ \Rightarrow \tan 300^\circ = \tan \left( {360^\circ - 60^\circ } \right)$.
Since tangent is a periodic function with its period at multiples of $\pi $(pie), we can write the above expression in RHS as:
$ \Rightarrow \tan 300^\circ = \tan \left( {360^\circ - 60^\circ } \right) = \tan \left( { - 60^\circ } \right)$
Now since we know that negative acute angles represent the angle from the fourth quadrant and the tangent function is negative in this quadrant. This gives us an identity: $\tan \left( { - x} \right) = - \tan x$.
Using that, we can write:
$ \Rightarrow \tan 300^\circ = \tan \left( { - 60^\circ } \right) = - \tan \left( {60^\circ } \right)$
According to the definition of the tangent, it can be expressed as the ratio of sine and cosine
$ \Rightarrow \tan 60^\circ = \dfrac{{\sin 60^\circ }}{{\cos 60^\circ }} = \dfrac{{\dfrac{{\sqrt 3 }}{2}}}{{\dfrac{1}{2}}} = \sqrt 3 $.
Therefore, the required value will be:
$ \Rightarrow \tan 300^\circ = - \tan \left( {60^\circ } \right) = - \sqrt 3 $

Thus, the value for the expression $\tan \left( {\dfrac{{5\pi }}{3}} \right)$ is $ - \sqrt 3 $ or $ - 1.732$.

Note: The coordinate plane is divided into four equal parts by x-axis and y-axis and each of these sections is called quadrants. All three trigonometric ratios (sin, cos, and tan) are positive in Quadrant I, Sine only is positive in Quadrant II, Tangent only is positive in Quadrant III and Cosine only is positive in Quadrant IV. An alternate approach can be to represent the given angle as $\dfrac{{5\pi }}{3} = \pi + \dfrac{{2\pi }}{3}$ . Since the tangent repeats itself at every $\pi $ angle, this can be used to find the value for $\tan \left( {\pi + \dfrac{{2\pi }}{3}} \right)$.