Question

Find the exact value of each expression. Do not use a calculator.

$2\sin \frac{\pi }{3}-3\tan \frac{\pi }{6}$

Short answer

The exact value of the given expression is 0.

Step-by-step solution

Step 1. Given information.

The given expression is:

$2\sin \frac{\pi }{3}-3\tan \frac{\pi }{6}$

Step 2. Determine the exact value.

In the trigonometric table $\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}$and $\tan \frac{\pi }{6}=\frac{1}{\sqrt{3}}$. Substitute these values in the given expression.

$\begin{array}{rcl}2\sin \frac{\pi }{3}-3\tan \frac{\pi }{6}& =& 2\left(\frac{\sqrt{3}}{2}\right)-3\left(\frac{1}{\sqrt{3}}\right)\\ & =& \sqrt{3}-\sqrt{3}\\ & =& 0\end{array}$

Step 3. Conclusion.

The exact value of the given expression is 0.