Question

Evaluating an Expression

Evaluate the expression without using a calculator

$\mathit{s}\mathit{i}\mathit{n}{\mathbf{30}}^{\mathbf{°}}\mathit{c}\mathit{o}\mathit{s}{\mathbf{60}}^{\mathbf{°}}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}{\mathbf{60}}^{\mathbf{°}}\mathit{c}\mathit{o}\mathit{s}{\mathbf{30}}^{\mathbf{°}}$

Short answer

The answer of the expression is 1.

Step-by-step solution

Step 1. Given information

A trigonometric expression is given as $\sin {30}^{°}\cos {60}^{°}+\sin {60}^{°}\cos {30}^{°}$.

Step 2. Concept used

Use trigonometric ratios to simplify the equation. Replace the values of the ratios in the expression and do the computation. Also use trigonometric identities for simplification.

Step 3. Calculation

The given expression is $\sin {30}^{°}\cos {60}^{°}+\sin {60}^{°}\cos {30}^{°}$, substitute the values as-

$\begin{array}{l}\because \sin {30}^{°}=\frac{1}{2}\&\cos {60}^{°}=\frac{1}{2}\\ \because \sin {60}^{°}=\frac{\sqrt{3}}{2}\&\cos {30}^{°}=\frac{\sqrt{3}}{2}\\ =\left(\frac{1}{2}\times \frac{1}{2}\right)+\left(\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{2}\right)\\ =\frac{1}{4}+\frac{3}{4}\\ =\frac{4}{4}\\ =1\end{array}$