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Question
Evaluate sin \[\left( \tan^{- 1} \frac{3}{4} \right)\]
Solution
We know that
\[\tan^{- 1} x = \sin^{- 1} \frac{x}{\sqrt{1 + x^2}}\]
\[\therefore \sin\left( \tan^{- 1} \frac{3}{4} \right) = \sin\left\{ \sin^{- 1} \left( \frac{\frac{3}{4}}{\sqrt{1 + \frac{9}{16}}} \right) \right\}\]
\[ = \sin\left\{ \sin^{- 1} \left( \frac{\frac{3}{4}}{\frac{5}{4}} \right) \right\}\]
\[ = \sin\left( \sin^{- 1} \frac{3}{5} \right)\]
\[ = \frac{3}{5} \left[ \because \sin\left( \sin^{- 1} x \right) = x \right]\]
∴ \[\sin\left( \tan^{- 1} \frac{3}{4} \right) = \frac{3}{5}\]
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