Question

Find the value of $\tan {\displaystyle \frac{3\pi }{4}}$.

Answer 1

Hint: Angle ${\displaystyle \frac{3\pi }{4}}$ is an obtuse angle and is located in the 2nd quadrant. Use a trigonometric quadrant rule to find the value.

Complete step by step solution:
We can write $\tan {\displaystyle \frac{3\pi }{4}}$ = $\tan {\displaystyle \frac{\left(4-1\right)\pi }{4}}$
=> $\tan {\displaystyle \frac{3\pi }{4}}$= $\tan {\displaystyle \frac{\left(4\pi -\pi \right)}{4}}$
=> $\tan {\displaystyle \frac{3\pi }{4}}$= $\tan \left({\displaystyle \frac{4\pi }{4}}-{\displaystyle \frac{\pi }{4}}\right)$
Now in $\tan \left({\displaystyle \frac{4\pi }{4}}-{\displaystyle \frac{\pi }{4}}\right)$ we can cancel 4 from numerator and denominator from ${\sin }^{-1}\theta$
We get, $\tan {\displaystyle \frac{3\pi }{4}}$= $\tan \left(\pi -{\displaystyle \frac{\pi }{4}}\right)$ ……equation (1)
As we know from trigonometric identity
$\tan \left(\pi -x\right)$ = -$\tan x$
So we can write $\tan \left(\pi -{\displaystyle \frac{\pi }{4}}\right)$ = -$\tan {\displaystyle \frac{\pi }{4}}$……equation (2)
So putting equation (2) in equation (1) we get
$\tan {\displaystyle \frac{3\pi }{4}}$ = -$\tan {\displaystyle \frac{\pi }{4}}$
And we know, $\tan {\displaystyle \frac{\pi }{4}}$= 1

So $\tan {\displaystyle \frac{3\pi }{4}}$= -1

Note: Proof for $\tan \left(\pi -x\right)$
We know $\tan x$= ${\displaystyle \frac{\sin x}{\cos x}}$
So, $\tan \left(\pi -x\right)$ = ${\displaystyle \frac{\sin (\pi -x)}{\cos (\pi -x)}}$
And we know, $\sin \left(\pi -x\right)$ = $\sin x$
And $$\cos \left(\pi -x\right)$$ = -$\cos x$
So, $\tan \left(\pi -x\right)$ = ${\displaystyle \frac{\sin x}{-\cos x}}$
$\tan \left(\pi -x\right)$ = - $\tan x$.