Question

Given cot \(\theta\) = 3, the value of cos \(\theta\) is :

• A. \(\frac{1}{3}\)
• B. \(\frac{1}{\sqrt{10}}\)
• C. \(\frac{3}{\sqrt{10}}\)
• D. \(\frac{\sqrt{10}}{3}\)

Hint:

Alternatively, use the identity \(\text{cosec}^2 \theta = 1 + \cot^2 \theta\).
\(\text{cosec}^2 \theta = 1 + 3^2 = 10 \implies \sin \theta = \frac{1}{\sqrt{10}}\).
Then, \(\cos \theta = \cot \theta \times \sin \theta = 3 \times \frac{1}{\sqrt{10}} = \frac{3}{\sqrt{10}}\).

Answer 1

The Correct Option is C

To find the value of \( \cos \theta \) given \( \cot \theta = 3 \), we start by recalling the trigonometric identity related to cotangent:

• We know that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Given \( \cot \theta = 3 \), we have: \[\frac{\cos \theta}{\sin \theta} = 3\]
• This equation implies: \[\cos \theta = 3 \sin \theta\]

Next, we use the Pythagorean identity:

• \(\sin^2 \theta + \cos^2 \theta = 1\)

Substitute \( \cos \theta = 3 \sin \theta \) into the Pythagorean identity:

• \[\sin^2 \theta + (3 \sin \theta)^2 = 1\]
• Simplifying, we get: \[\sin^2 \theta + 9 \sin^2 \theta = 1\]

Combine the terms:

• \[10 \sin^2 \theta = 1\]

Solve for \( \sin^2 \theta \):

• \[\sin^2 \theta = \frac{1}{10}\]

Taking the square root on both sides, we determine \( \sin \theta \):

• \[\sin \theta = \frac{1}{\sqrt{10}}\]

Substitute back into \( \cos \theta = 3 \sin \theta \):

• \[\cos \theta = 3 \times \frac{1}{\sqrt{10}} = \frac{3}{\sqrt{10}}\]

Therefore, the correct answer is \(\frac{3}{\sqrt{10}}\).

This matches the provided correct answer, thus confirming our solution.