Question

Find the exact value of each expression.

$\sin \left({\sin }^{-1}\frac{\sqrt{3}}{2}+{\cos }^{-1}1\right)$

Short answer

The value of the expression is$\frac{\sqrt{3}}{2}$

Step-by-step solution

Step 1. Given information

The given expression is$\sin \left({\sin }^{-1}\frac{\sqrt{3}}{2}+{\cos }^{-1}1\right)$

Step 2. Solve the expression

Let $\alpha ={\sin }^{-1}\frac{\sqrt{3}}{2}$and $\beta ={\cos }^{-1}1$

So,

$\sin \alpha =\frac{\sqrt{3}}{2},-\frac{\mathrm{\pi }}{2}\le \alpha \le \frac{\mathrm{\pi }}{2}$and $\cos \beta =1,0\le \beta \le \mathrm{\pi }$

$$\begin{gathered} \cos \alpha =\sqrt{1-{\sin }^2\alpha } \\ \cos \alpha =\sqrt{1-{\left(\frac{\sqrt{3}}{2}\right)}^2} \\ \cos \alpha =\frac{1}{2} \end{gathered}$$

$$\begin{gathered} \sin \beta =\sqrt{1-{\cos }^2\beta } \\ \sin \beta =\sqrt{1-{\left(1\right)}^2} \\ \sin \beta =0 \end{gathered}$$

Now,

$$\begin{gathered} \sin \left(\alpha +\beta \right) \\ =\sin \alpha ·\cos \beta +\cos \alpha ·\sin \beta \\ =\frac{\sqrt{3}}{2}·1+\frac{1}{2}·0 \\ =\frac{\sqrt{3}}{2} \end{gathered}$$