Question

Evaluating an Expression

Evaluate the expression without using a calculator

${\left(sin\frac{\pi }{3}tan\frac{\pi }{6}+csc\frac{\pi }{4}\right)}^{\mathbf{2}}$

Short answer

The answer of the expression is$\frac{\mathbf{9}\mathbf{+}\mathbf{4}\sqrt{\mathbf{2}}}{\mathbf{4}}$

Step-by-step solution

Step 1. Given information

A trigonometric expression is given as${\left(\sin \frac{\pi }{3}\tan \frac{\pi }{6}+\csc \frac{\pi }{4}\right)}^2$

Step 2. Concept used

Substitute the values of trigonometric ratios in the given expression. After that, simplify it with the help of trigonometric identities.

Step 3. Calculation

The given expression is ${\left(\sin \frac{\pi }{3}\tan \frac{\pi }{6}+\csc \frac{\pi }{4}\right)}^2$, substitute the values as-

We know that,

$\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2},\tan \frac{\pi }{6}=\frac{1}{\sqrt{3}}\&\csc \frac{\pi }{4}=\sqrt{2}$

Now simplify the expression,

$\begin{array}{l}={\left(\frac{\sqrt{3}}{2}\times \frac{1}{\sqrt{3}}+\sqrt{2}\right)}^2\\ ={(\frac{1}{2}+\sqrt{2})}^2={\left(\frac{1+2\sqrt{2}}{2}\right)}^2\\ =\frac{1+8+4\sqrt{2}}{4}\\ =\frac{9+4\sqrt{2}}{4}\end{array}$