Question

How do you evaluate $\tan \left( 17\pi \right)$ ?

Answer 1

Hint: We have been given the trigonometric function, tangent of $17\pi $ whose value is to be calculated. We shall first break down the given angle as the sum of two known conventional angles, that is, $17\pi =16\pi +\pi $. Then, we shall apply the formula of the sum of angles of tangent function which is given as $\tan \left( x+y \right)=\dfrac{\tan x+\tan y}{1-\tan x.\tan y}$. Further, after substituting all the necessary values, we shall calculate the value of $\tan \left( 17\pi \right)$.

Complete step by step solution:
Given the trigonometric function tangent of angle $17\pi $or $\tan \left( 17\pi \right)$.
Since the given angle, $17\pi $ is not a conventional angle whose values are usually memorized, thus we shall break it into the two conventional angles whose values we know.
Hence, $17\pi $ can be expressed as $17\pi =16\pi +\pi $.
On substituting this value in the tangent function, we have
$\Rightarrow \tan \left( 17\pi \right)=\tan \left( 16\pi +\pi \right)$
Here, we shall use the property of tangent of sum of two angles in which the sum of two angles can be expressed in terms of those individual two angles, that is, $\tan \left( x+y \right)=\dfrac{\tan x+\tan y}{1-\tan x.\tan y}$.
We have $x=16\pi $ and $y=\pi $.
$\Rightarrow \tan \left( 17\pi \right)=\dfrac{\tan 16\pi +\tan \pi }{1-\tan 16\pi .\tan \pi }$
Now, we know that the period of tangent function is $\pi $, thus $\tan \left( 16\pi \right)$ can be written as
$\tan 16\pi =\tan 8\left( 2\pi \right)$
$\Rightarrow \tan 16\pi =8\tan \left( 2\pi \right)$
And since $\tan 2\pi =0$, therefore
$\Rightarrow \tan 16\pi =8\left( 0 \right)$
$\Rightarrow \tan 16\pi =0$
Also, $\tan \pi =0$. Substituting these values, we get
$\Rightarrow \tan \left( 17\pi \right)=\dfrac{0+0}{1-0.\left( 0 \right)}$
$\Rightarrow \tan \left( 17\pi \right)=\dfrac{0}{1}$
$\Rightarrow \tan \left( 17\pi \right)=0$
Therefore, $\tan \left( 17\pi \right)$ is evaluated to be equal to 0.

Note: Another method of solving this problem was by writing the tangent function as $\dfrac{\sin x}{\cos x}$. We would then write the given angle as the sum of two angles and solve for sine and cosine functions. Also, another way of obtaining the value of given trigonometric quantity was by writing $\tan \left( 17\pi \right)=17\tan \pi $ as the period of tangent function is $\pi $ and then simply substituting the value of $\tan \pi =0$.