Question

If $\cos x=\frac{3}{5}$ where $x\in (\frac{3\pi }{2},2\pi )$, then the value of $\sin 4x$ is

• A. $\frac{336}{625}$
• B. $\frac{144}{625}$
• C. $\frac{224}{625}$
• D. $\frac{336}{475}$

Answer 1

The correct option is A $\frac{336}{625}$

$\cos x=\frac{3}{5}$
As $x\in (\frac{3\pi }{2},2\pi )$
$\Rightarrow \sin x=-\frac{4}{5}$

Now,
$\sin 4x=2\sin 2x\cos 2x=4\sin x\cos x\cos 2x=4\sin x\cos x(2{\cos }^{2}x-1)=4\times (-\frac{4}{5})\times \frac{3}{5}[2{\left(\frac{3}{5}\right)}^2-1]=-\frac{48}{25}\times [\frac{18}{25}-1]=\frac{48}{25}\times \frac{7}{25}=\frac{336}{625}$