2 ^ -1 (-3 5 ) is equal to: 1. 6 25 2. 24 25 3. 4 5 4. -24 25 - Vidyadip

Question

$\sin\Big\{2\cos^{-1}\Big(\frac{-3}{5}\Big)\Big\}$ is equal to:

$\frac{6}{25}$
$\frac{24}{25}$
$\frac{4}{5}$
$-\frac{24}{25}$

✓

Answer

$-\frac{24}{25}$

Solution:
Let $\cos^{-1}\Big(-\frac{3}{5}\Big)=\text{x},0\leq\text{x}\leq\pi$
Then, $\cos\text{x}=-\frac{3}{5}$
$\therefore\ \sin\text{x}=\sqrt{1-\cos^2\text{x}}=\sqrt{1-\Big(-\frac{3}{5}\Big)^2}=\sqrt{\frac{16}{25}}=\frac{4}{5}$
Now,
$\sin\Big\{2\cos^{-1}\Big(\frac{-3}{5}\Big)\Big\}=\sin(2\text{x})$
$=2\sin\text{x}\cos\text{x}$
$=2\times\frac{4}{5}\times\frac{-3}{5}$
$=-\frac{24}{25}$

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