Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function

The pattern displayed in this problem is $\sin \left(\alpha +\beta \right).$ Let $\alpha ={\cos }^{\mathrm{-1}}\frac{1}{2}$ and $\beta ={\sin }^{\mathrm{-1}}\frac{3}{5}.$ Then we can write

We will use the Pythagorean Identities to find $\sin \alpha$ and $\cos \beta .$

Using the sum formula for sine,