Question

If $\sin \theta =-\frac{4}{5},\pi <\theta <\frac{3\pi }{2}$, then find
i) $\sin 2\theta$
ii) $\cos 2\theta$
iii) $\tan 2\theta$

Answer 1

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

$\cos \theta =-\sqrt{1-{\left(\frac{-4}{5}\right)}^2}=\frac{-3}{5},\tan \theta =\frac{4}{3}$

$\left(i\right)\sin 2\theta =2\sin \theta \cos \theta =2\times \left(\frac{-4}{3}\right)\left(\frac{-3}{5}\right)=\frac{24}{25}$

$\left(ii\right)\cos 2\theta =2{\cos }^2\theta -1=2{\left(\frac{-3}{5}\right)}^2-1=\frac{18}{25}-1=\frac{-7}{25}$

$\left(iii\right)\tan 2\theta =\frac{\sin 2\theta }{\cos 2\theta }=\frac{24/25}{-7/25}=\frac{-24}{7}$