Question

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

$3\csc \frac{\pi }{3}+\cot \frac{\pi }{4}$

Short answer

The exact value of the given expression is $2\sqrt{3}+1$.

Step-by-step solution

Step 1. Given information.

The given expression is:

$3\csc \frac{\pi }{3}+\cot \frac{\pi }{4}$

Step 2. Determine the exact value.

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

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

Step 3. Conclusion.

The exact value of the given expression is $2\sqrt{3}+1$.