Question:
If $\sin A=\frac{1}{2}$ then the value of $\cot A$ is
(a) $\sqrt{3}$
(b) $\frac{1}{\sqrt{3}}$
(c) $\frac{\sqrt{3}}{1}$
(d) 1
Solution:
(a) Given, $\sin A=\frac{1}{2}$
$\therefore \quad \cos A=\sqrt{1-\sin ^{2} A}=\sqrt{1-\left(\frac{1}{2}\right)^{2}}$
$=\sqrt{1-\frac{1}{4}}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$ $\left[\because \sin ^{2} A+\cos ^{2}=1 \Rightarrow \cos A=\sqrt{1-\sin ^{2} A}\right]$
Now, $\cot A=\frac{\cos A}{\sin A}=\frac{\frac{\sqrt{3}}{2}}{1}=\sqrt{3}$