Question

How do you evaluate $\tan \left( \dfrac{7\pi }{4} \right)$ ?

Answer 1

Hint: We have been given the trigonometric function, tangent of $\dfrac{7\pi }{4}$ whose value is to be calculated. We shall first break down the given angle as the difference of two known conventional angles, that is, $\dfrac{7\pi }{4}=2\pi -\dfrac{\pi }{4}$. Then, we shall apply the formula of the difference 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 $\tan \left( \dfrac{7\pi }{4} \right)$.

Complete step by step solution:
Given the trigonometric function tangent of angle $\dfrac{7\pi }{4}$ or $\tan \left( \dfrac{7\pi }{4} \right)$.
Since the given angle, $\dfrac{7\pi }{4}$ 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, $\dfrac{7\pi }{4}$ can be expressed as $\dfrac{7\pi }{4}=2\pi -\dfrac{\pi }{4}$.
On substituting this value in the tangent function, we have
$\Rightarrow \tan \left( \dfrac{7\pi }{4} \right)=\tan \left( 2\pi -\dfrac{\pi }{4} \right)$
Here, we shall use the property of tangent of difference of two angles in which the difference 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=2\pi $ and $y=\dfrac{\pi }{4}$.
$\Rightarrow \tan \left( \dfrac{7\pi }{4} \right)=\dfrac{\tan 2\pi -\tan \dfrac{\pi }{4}}{1+\tan 2\pi .\tan \dfrac{\pi }{4}}$
Now, we know that $\tan 2\pi =0$ and $\tan \dfrac{\pi }{4}=1$. Substituting these values, we get
$\Rightarrow \tan \left( \dfrac{7\pi }{4} \right)=\dfrac{0-1}{1+0.\left( 1 \right)}$
$\Rightarrow \tan \left( \dfrac{7\pi }{4} \right)=\dfrac{-1}{1}$
$\Rightarrow \tan \left( \dfrac{7\pi }{4} \right)=-1$
Therefore, $\tan \left( \dfrac{7\pi }{4} \right)$ is evaluated to be equal to $-1$.

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 difference of two conventional angles as $\dfrac{7\pi }{4}=2\pi -\dfrac{\pi }{4}$ for both sine function as well as the cosine functions and apply the identity of difference of angles of sine function and cosine function as $\sin \left( a-b \right)=\sin a\cos b-\cos a\sin b$ and $\cos \left( a-b \right)=\cos a\cos b+\sin a\sin b$ respectively.