Question : If $\cos A=\frac{1}{11}$, then find the value of cot A.
Option 1: $\frac{2 \sqrt{30}}{11}$
Option 2: $2 \sqrt{30}$
Option 3: $\frac{1}{2 \sqrt{30}}$
Option 4: $\frac{11}{2 \sqrt{30}}$
Correct Answer: $\frac{1}{2 \sqrt{30}}$
Solution :
Given,
$\cos A=\frac{1}{11}$
We know, $\sin^2\theta+\cos^2\theta=1$
⇒ $\sin^2A+\cos^2A=1$
⇒ $\sin^2A+\frac{1}{11^2}=1$
⇒ $\sin^2A=1-\frac{1}{121}$
⇒ $\sin^2A=\frac{121-1}{121}$
⇒ $\sin^2 A=\frac{120}{121}$
⇒ $\sin A=\sqrt{\frac{120}{121}}$
⇒ $\sin A=\frac{2\sqrt{30}}{11}$
⇒ $\cot A = \frac{\cos A}{\sin A}$
⇒ $\cot A = \frac{\frac{1}{11}}{\frac{2\sqrt{30}}{11}}$
⇒ $\cot A=\frac{1}{2\sqrt{30}}$
Hence, the correct answer is $\frac{1}{2\sqrt{30}}$.