Question

If cosecx =-2 and pie<x<3pie/2
then find
the values of
(i)sin 2x
(iii) sin 3x
(ii) tan 2x
(iv) sin 4X​

Answer 1

$\pi\ \textless\ x\ \textless\ \dfrac {3\pi}{2}$ means $x$ is nothing but in third quadrant. So,

$\quad$

$\csc x=-2=-\csc\left (\dfrac {\pi}{6}\right)\quad\!\implies\quad\!x=\dfrac {7\pi}{6}$

$\quad$

Then,

$\quad$

$\sin(2x)=\sin\left (\dfrac {14\pi}{6}\right)=\sin\left (\dfrac {\pi}{3}\right)=\bf{\dfrac {\sqrt 3}{2}}\\\\\\\sin (3x)=\sin\left (\dfrac {21\pi}{6}\right)=\sin\left (\dfrac {3\pi}{2}\right)=\bf{-1}\\\\\\\tan(2x)=\tan\left (\dfrac {14\pi}{6}\right)=\tan\left (\dfrac {\pi}{3}\right)=\bf{\sqrt 3}\\\\\\\sin (4x)=\sin\left (\dfrac {28\pi}{6}\right)=\sin\left (\dfrac {2\pi}{3}\right)=\bf{\dfrac {\sqrt 3}{2}}$