Question

if $\sin \theta =-\frac{4}{5},\pi <\theta <\frac{3\pi }{2},$ then find
1. $\sin 2\theta$
2. $\cos 2\theta$
3. $\tan 2\theta$

Answer 1

Solution :-

Given $sin\theta =\frac{-4}{5}$ and $\pi <\theta <\frac{3\pi }{2}$

then

$1.sin2\theta =2sin\theta .cos\theta$

$cos\theta =-\frac{\sqrt{5^2-4^2}}{5}=-\frac{3}{5}$ $\because \pi <\theta <\frac{3\pi }{2}$

$\therefore sin2\theta =2\times \frac{-4}{5}\times \frac{-3}{5}=\frac{24}{25}$

$2.cos2\theta =co{s}^2\theta -si{n}^2\theta$

$=(\frac{-3}{5}{)}^2-(\frac{-4}{5}{)}^2$

$=\frac{9}{25}-\frac{16}{25}$

$=\frac{-7}{25}$

$3.tan2\theta =\frac{sin2\theta }{tan2\theta }=\frac{24/25}{-7/25}$

$=\frac{-24}{7}$