Question

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

$2\sec \frac{\pi }{4}+4\cot \frac{\pi }{3}$

Short answer

The exact value of the given expression is $2\sqrt{2}+\frac{4\sqrt{3}}{3}$.

Step-by-step solution

Step 1. Given information.

The given expression is:

$2\sec \frac{\pi }{4}+4\cot \frac{\pi }{3}$

Step 2. Determine the exact value.

In the trigonometric table $\sec \frac{\pi }{4}=\sqrt{2}$and $\cot \frac{\pi }{3}=\frac{1}{\sqrt{3}}$. Substitute these values in the given expression.

$\begin{array}{rcl}2\sec \frac{\pi }{4}+4\cot \frac{\pi }{3}& =& 2\left(\sqrt{2}\right)+4\left(\frac{1}{\sqrt{3}}\right)\\ & =& 2\sqrt{2}+\frac{4}{\sqrt{3}}\\ & =& 2\sqrt{2}+\frac{4\sqrt{3}}{3}\end{array}$

Step 3. Conclusion.

The exact value of the given expression is $2\sqrt{2}+\frac{4\sqrt{3}}{3}$.