Mark the tick against the correct answer in the following:

Question:

Mark the tick against the correct answer in the following:

$\sin \left[2 \sin ^{-1} \frac{4}{5}\right]$

A. $\frac{12}{25}$

B. $\frac{16}{25}$

C. $\frac{24}{25}$

D. None of these

 

Solution:

To Find: The value of $\sin \left(2 \sin ^{-1} \frac{4}{5}\right)$

Let, $x=\sin ^{-1} \frac{4}{5}$

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

We know that, $\cos x=\sqrt{1-\sin ^{2} x}$

$=\sqrt{1-\left(\frac{4}{5}\right)^{2}}$

$=\frac{3}{5}$

Now since, $x=\sin ^{-1} \frac{4}{5}$, hence $\sin \left(2 \sin ^{-1} \frac{4}{5}\right)$ becomes $\sin (2 x)$

Here, $\sin (2 x)=2 \sin x \cos x$

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

$=\frac{24}{25}$

 

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