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 trigonometric identities:
\( \csc^2 \theta = 1 + \cot^2 \theta = 1 + 9 = 10 \implies \sin\theta = \frac{1}{\sqrt{10}} \).
Since \( \cos\theta = \cot\theta \cdot \sin\theta \), we get \( \cos\theta = 3 \cdot \frac{1}{\sqrt{10}} = \frac{3}{\sqrt{10}} \).
Identity methods are often faster than drawing triangles!

Answer 1

The Correct Option is C

Step 1: Use the identity connecting cosec and cot.
Instead of drawing a right triangle, we can go straight from $\cot\theta$ to $\cos\theta$ using the identity
\[ \csc^2\theta = 1 + \cot^2\theta \]
which holds for any angle $\theta$.

Step 2: Substitute the given value.
We are given $\cot\theta = 3$, so:
\[ \csc^2\theta = 1 + 3^2 = 1 + 9 = 10 \implies \csc\theta = \sqrt{10} \]
so $\sin\theta = \dfrac{1}{\sqrt{10}}$, taking $\theta$ as acute so every ratio stays positive.

Step 3: Recover cos theta from the definition of cot.
By definition, $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$, so $\cos\theta = \cot\theta \times \sin\theta$:
\[ \cos\theta = 3 \times \frac{1}{\sqrt{10}} = \frac{3}{\sqrt{10}} \]

Final Answer:
This matches option (C).
\[ \boxed{\cos\theta = \frac{3}{\sqrt{10}}} \]