Question

Find the values of $\sin ({\cos }^{-1}\frac{3}{5}+{\cos }^{-1}\frac{12}{13})$.

Answer 1

$Sin.\left({\cos }^{-1}\left(\frac{3}{5}\right)+{\cos }^{-1}\frac{12}{13}\right)Let{\cos }^{-1}\frac{3}{5}=\theta ,{\cos }^{-1}\frac{12}{13}=0\Rightarrow \cos \theta =\frac{3}{5}\Rightarrow \cos 0=\frac{12}{13}\therefore \sin \theta =\sqrt{1-{\left(\frac{3}{5}\right)}^2}\Rightarrow \sin 0=\sqrt{1-{\left(\frac{12}{13}\right)}^2}\Rightarrow \sin \theta =\sqrt{\frac{25-9}{25}}\Rightarrow \sin 0=\sqrt{\frac{169-144}{169}}\sin \theta =\frac{4}{5}\sin 0=\frac{5}{13}\Rightarrow \sin \left(\theta +0\right)=\sin \theta .\cos 0+\sin 0.\cos \theta \therefore \sin \left(\theta +0\right)=\frac{4}{5}.\frac{12}{13}+\frac{5}{13}\times \frac{3}{5}=\frac{48}{65}+\frac{15}{65}=\frac{63}{65}$