Evaluate ( ^ -1 3 4 ). - Vidyadip

Question

Evaluate $\sin\Big(\tan^{-1}\frac{3}{4}\Big).$

✓

Answer

We know that
$\tan^{-1}\text{x}=\sin^{-1}\frac{\text{x}}{\sqrt{1+\text{x}^2}}$
$\therefore\ \sin\Big(\tan^{-1}\frac{3}{4}\Big)=\sin\Bigg\{\sin^{-1}\Bigg(\frac{\frac{3}{4}}{\sqrt{1+\frac{9}{10}}}\Bigg)\Bigg\}$
$=\sin\Bigg\{\sin^{-1}\Bigg(\frac{\frac{3}{4}}{\frac{5}{4}}\Bigg)\Bigg\}$
$=\sin\Big(\sin^{-1}\frac{3}{5}\Big)$
$=\frac{3}{5}$ $\big[\because\ \sin\big(\sin^{-1}\text{x}\big)=\text{x}\big]$
$\therefore\ \Big(\tan^{-1}\frac{3}{4}\Big)=\frac{3}{5}$

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