Question

Find the exact value of each expression.

$\tan \left({\sin }^{-1}\frac{3}{5}+\frac{\mathrm{\pi }}{6}\right)$

Short answer

The value of the expression is$\frac{48+25\sqrt{3}}{39}$

Step-by-step solution

Step 1. Given information

The given expression is$\tan \left({\sin }^{-1}\frac{3}{5}+\frac{\mathrm{\pi }}{6}\right)$

Step 2. Solve the expression

Let $\alpha ={\sin }^{-1}\frac{3}{5}$and $\beta =\frac{\mathrm{\pi }}{6}$

So,

$$\begin{gathered} \tan \alpha =\frac{3}{4} \\ \tan \beta =\frac{1}{\sqrt{3}} \end{gathered}$$

Now,

localid="1646561606613" $$\begin{gathered} \tan \left(\alpha +\beta \right) \\ =\frac{\tan \alpha +\tan \beta }{1-\tan \alpha ·\tan \beta } \\ =\frac{{\displaystyle \frac{3}{4}}+{\displaystyle \frac{1}{\sqrt{3}}}}{1-{\displaystyle \frac{3}{4}}·{\displaystyle \frac{1}{\sqrt{3}}}} \\ =\frac{3\sqrt{3}+4}{4\sqrt{3}-3} \\ =\frac{3\sqrt{3}+4}{4\sqrt{3}-3}\times \frac{4\sqrt{3}+3}{4\sqrt{3}+3} \\ =\frac{48+25\sqrt{3}}{39} \end{gathered}$$