Evaluate the following: ( ^ -1 24 7 ) - Vidyadip

Question

Evaluate the following:
$\sin\Big(\tan^{-1}\frac{24}{7}\Big)$

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Answer

$\sin\Big(\tan^{-1}\frac{24}{7}\Big)$
$=\sin\begin{pmatrix}\sin^{-1}\frac{\frac{24}{7}}{\sqrt{1+\big(\frac{24}{7}\big)^2}}\end{pmatrix}$ $\Big[{\therefore\ \tan^{-1}}\text{x}=\frac{\text{x}}{\sqrt{1+\text{x}^2}}\Big]$
$=\sin\begin{pmatrix}\sin^{-1}\frac{\frac{24}{7}}{\sqrt{1+\frac{576}{49}}}\end{pmatrix}$
$=\sin\begin{pmatrix}\sin^{-1}\frac{\frac{24}{7}}{\sqrt{\frac{625}{49}}}\end{pmatrix}$
$=\sin\begin{pmatrix}\sin^{-1}\frac{\frac{24}{7}}{\frac{25}{7}}\end{pmatrix}$
$=\frac{24}{25}$

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