Question

Find the value of $\sin \left[2{\cos }^{-1}(-\frac{3}{5})\right]$

• A. $\frac{6}{25}$
• B. $\frac{24}{25}$
• C. $\frac{4}{25}$
• D. $-\frac{24}{25}$

Answer 1

The correct option is D $-\frac{24}{25}$

$\sin \left[2{\cos }^{-1}(-\frac{3}{5})\right]$

Let ${\cos }^{-1}(-\frac{3}{5})=a$

$\Rightarrow \cos a=-\frac{3}{5}$

$\Rightarrow {\cos }^{2}a=\frac{9}{25}$

$\Rightarrow {\sin }^{2}a=1-{\cos }^{2}a=1-\frac{9}{25}=\frac{25-9}{25}=\frac{16}{25}$

$\Rightarrow \sin a=\frac{4}{5}$

Now,

$\sin \left[2{\cos }^{-1}(-\frac{3}{5})\right]$$=\sin 2a$

$=2\sin a\cos a$

$=2\times \frac{4}{5}\times -\frac{3}{5}$

$=-\frac{24}{25}$