Evaluate each of the following: `sin(sin^(-1)(7/(25)))` (ii) `sin(cos^(-1)(5/(13)))` (iii) `sin(tan^(-1)((24)/7))`

Explanation

(i) `\sin (\sin ^{-1} \frac{7}{25})=\frac{7}{25}`

(ii) `\sin (\cos ^{-1} \frac{5}{13})=\sin [\sin ^{-1} \sqrt{1-(\frac{5}{13})^{2}}] \quad[\because \cos ^{-1} x=\sin ^{-1} \sqrt{1-x^{2}}]`

`=\sin [\sin ^{-1}(\sqrt{1-\frac{25}{169}})]`

`=\sin [\sin ^{-1}(\sqrt{\frac{144}{169}})]`

`=\sin [\sin ^{-1} \frac{12}{13}]`

`=\frac{12}{13}`


(iii) `\sin (\tan ^{-1} \frac{24}{7})=\sin (\sin ^{-1} \frac{\frac{24}{7}}{\sqrt{1+(\frac{24}{7})^{2}}}) \quad[\because \tan ^{-1} x=\frac{x}{\sqrt{1+x^{2}}}]`

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