Question

Find the exact value of the expression:

$\tan \left(\frac{7\mathrm{\pi }}{12}\right)$

Short answer

$\left(\frac{1+\sqrt{3}}{1-\sqrt{3}}\right)$

Step-by-step solution

Step 1. Given information 

We have been given an expression $\tan \left(\frac{7\mathrm{\pi }}{12}\right)$.

We have to find its exact value.

Step 2. Simplify the expression

We can also write the expression as:

$$\begin{gathered} \tan \left(\frac{7\mathrm{\pi }}{12}\right)=\tan \left(\left(\frac{3\mathrm{\pi }}{12}\right)+\left(\frac{4\mathrm{\pi }}{12}\right)\right) \\ =\tan \left(\left(\frac{\mathrm{\pi }}{4}\right)+\left(\frac{\mathrm{\pi }}{3}\right)\right) \end{gathered}$$

Step 3.  Apply the compound angle formula

$\tan \left(\left(\frac{\mathrm{\pi }}{4}\right)+\left(\frac{\mathrm{\pi }}{3}\right)\right)$has the form of compound angle localid="1646468051241" $\tan \left(\alpha +\beta \right)$

Therefore, apply

$\tan \left(\alpha +\beta \right)=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }$

$\begin{array}{r}\frac{\tan \left(\frac{\pi }{4}\right)+\tan \left(\frac{\pi }{3}\right)}{1-\tan \left(\frac{\pi }{4}\right)\tan \left(\frac{\pi }{3}\right)}=\frac{1+\sqrt{3}}{1-\left(1\right(\sqrt{3}\left)\right)}\\ =\left(\frac{1+\sqrt{3}}{1-\sqrt{3}}\right)\end{array}$