Question

In the given equilateral triangle ABC, find the value of sin ${30}^{\circ }$ and cos ${30}^{\circ }$, without using tables.

AD is the perpendicular bisector of BC.

• A. $\frac{1}{2}$, $\frac{1}{2}$
• B. $\frac{1}{2}$, $\frac{\sqrt{3}}{2}$
• C. $\frac{1}{\sqrt{2}}$, $\sqrt{2}$
• D. $\frac{1}{\sqrt{3}}$, $\sqrt{3}$

Answer 1

The correct option is B

$\frac{1}{2}$, $\frac{\sqrt{3}}{2}$

Let each side of the equilateral triangle be a.

AD is the perpendicular bisector of BC. (which is also the median and angle bisector).

In $△$ADB

$A{D}^2$ = $A{B}^2-B{D}^2$

= $a^2-\frac{a^2}{4}$

= $\frac{3a^2}{4}$

$AD=\frac{\sqrt{3}a}{2}$

$sin30=\frac{BD}{AB}$ = $\frac{\left(\frac{a}{2}\right)}{a}$ = $\frac{1}{2}$

$cos{30}^{\circ }$ = $\frac{AD}{AB}$ = $\frac{\frac{\sqrt{3}a}{2}}{a}$ = $\frac{\sqrt{3}}{2}$