Question

Evaluating an Expression

Evaluate the expression without using a calculator

${\left(sin\frac{\pi }{3}cos\frac{\pi }{4}-sin\frac{\pi }{4}cos\frac{\pi }{3}\right)}^{\mathbf{2}}$

Short answer

The answer of the expression is$\frac{\mathbf{2}\mathbf{-}\sqrt{\mathbf{3}}}{\mathbf{4}}$

Step-by-step solution

Step 1. Given information

A trigonometric expression is given as ${\left(\sin \frac{\pi }{3}\cos \frac{\pi }{4}-\sin \frac{\pi }{4}\cos \frac{\pi }{3}\right)}^2$.

Step 2. Concept used

Use trigonometric ratios to simplify the issue. Substitute the ratio values in the formulation and calculate. Use trigonometric identities for even greater simplification.

Step 3. Calculation

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

We know that,

$\begin{array}{l}\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}\\ \cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}\\ \sin \frac{\pi }{4}=\frac{1}{\sqrt{2}}\\ \cos \frac{\pi }{3}=\frac{1}{2}\end{array}$

role="math" localid="1648542304835" $\begin{array}{l}{\left(\sin \frac{\pi }{3}\cos \frac{\pi }{4}-\sin \frac{\pi }{4}\cos \frac{\pi }{3}\right)}^2\\ ={(\frac{\sqrt{3}}{2}\times \frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\times \frac{1}{2})}^2\\ ={\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)}^2\\ =\frac{{\left(\sqrt{3}\right)}^2+1^2-2\left(1\right)\left(\sqrt{3}\right)}{{\left(2\sqrt{2}\right)}^2}\\ =\frac{3+1-2\sqrt{3}}{8}\\ =\frac{4-2\sqrt{3}}{8}\\ =\frac{2-\sqrt{3}}{4}\end{array}$