Question

If $\sin \left(A\right)=\frac{1}{2}$ , then the value of $\cot \left(A\right)$ is:

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

Answer 1

The correct option is A

$\sqrt{3}$

Explanation for the correct options:

Given : $\sin \left(A\right)=\frac{1}{2}$………………$\left(1\right)$

Since,

$\begin{array}{rcl}\cot \left(A\right)& =& \frac{1}{\tan \left(A\right)}\\ & =& \frac{\cos \left(A\right)}{\sin \left(A\right)}\end{array}$.

We know the identity:

$\begin{array}{rcl}{\sin }^2\left(\theta \right)+{\cos }^2\left(\theta \right)& =& 1\\ & \Rightarrow & \cos \left(\theta \right)=\sqrt{1-{\sin }^2\left(\theta \right)}\end{array}$

Now,

$\Rightarrow$ $\cos \left(A\right)=\sqrt{1-{\sin }^2\left(A\right)}$…………….$\left(2\right)$

From equation $\left(1\right)$ ,

$\begin{array}{rcl}& \Rightarrow & \cos \left(A\right)=\sqrt{1-\frac{1}{4}}\\ & =& \sqrt{\frac{3}{4}}\\ & =& \frac{\sqrt{3}}{2}\end{array}$

Since, $\begin{array}{rcl}\cot \left(A\right)& =& \frac{\cos \left(A\right)}{\sin \left(A\right)}\end{array}$

$\begin{array}{rcl}\cot \left(A\right)& =& \left(\frac{\sqrt{3}}{2}\right)\times 2\\ & =& \sqrt{3}\end{array}$

Hence, option $\left(A\right)$ is the correct option.