Question 6.6.5.9: finding the exact Value of an expression involving inverse t......
finding the exact Value of an expression involving inverse trigonometric functions
Find the exact value of: \sin\left(\cos^{-1}{\frac{1}{2}}+\sin^{-1}{\frac{3}{5}}\right)
We seek the sine of the sum of two angles, \alpha=\cos^{-1}{\frac{1}{2}} and \beta=\sin^{-1}{\frac{3}{5}}. Then
\cos\alpha={\frac{1}{2}}\quad\quad0\leq\alpha\leq\pi\quad{\mathrm{and}}\quad\sin\beta={\frac{3}{5}}\quad\quad-{\frac{\pi}{2}}\leq\beta\leq{\frac{\pi}{2}}Use Pythagorean Identities to obtain \sin \alpha and \cos \beta. Because \alpha \geq 0 {because 0\,\leq\,\alpha\ \leq\,\pi) and \cos\beta\geq0 (because -\frac{\pi}{2}\leq\beta\ \leq\frac{\pi}{2}), this means that
\sin\alpha={\sqrt{1-\cos^{2}\alpha}}={\sqrt{1-{\frac{1}{4}}}}={\sqrt{{\frac{3}{4}}}}={\frac{\sqrt{3}}{2}}\cos\beta={\sqrt{1-\sin^{2}\beta}}={\sqrt{1-{\frac{9}{25}}}}={\sqrt{\frac{16}{25}}}={\frac{4}{5}}
As a result,
\sin\left(\cos^{-1}{\frac{1}{2}}+\sin^{-1}{\frac{3}{5}}\right)=\sin\left(\alpha+\beta\right)=\sin\alpha\cos\beta+\cos\alpha\sin\beta={\frac{\sqrt{3}}{2}}\cdot{\frac{4}{5}}+{\frac{1}{2}}\cdot{\frac{3}{5}}={\frac{4{\sqrt{3}}+3}{10}}
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