Question

If \(\sin \theta = \frac{1}{\sqrt{11}}\text{, then } \cot \theta\) equals

• A. \(\frac{\sqrt{11}}{\sqrt{10}}\)
• B. \(\frac{\sqrt{10}}{\sqrt{11}}\)
• C. \(\sqrt{10}\)
• D. \(\sqrt{11}\)

Hint:

Using trigonometric identities is often faster:
\[ \cot^2 \theta = \csc^2 \theta - 1 \]
Since \(\csc \theta = \frac{1}{\sin \theta} = \sqrt{11}\):
\[ \cot^2 \theta = (\sqrt{11})^2 - 1 = 11 - 1 = 10 \]
\[ \cot \theta = \sqrt{10} \]
This identity-based approach helps avoid drawing triangles.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the value of \(\sin \theta = \frac{1}{\sqrt{11}}\).
We need to find the value of \(\cot \theta\).

Step 2: Key Formula or Approach:
In a right-angled triangle, trigonometric ratios are defined as:
\[ \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \]
\[ \cot \theta = \frac{\text{Adjacent Side}}{\text{Opposite Side}} \]
We can use Pythagoras' theorem to find the length of the adjacent side:
\[ \text{Hypotenuse}^2 = \text{Opposite Side}^2 + \text{Adjacent Side}^2 \]
Alternatively, we can use the fundamental trigonometric identity:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta} \]

Step 3: Detailed Explanation:

• Let us construct a right-angled triangle where the angle is \(\theta\).

• From the given ratio, we can write:
\[ \sin \theta = \frac{1}{\sqrt{11}} = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
Let the Opposite side be \(1k\) and the Hypotenuse be \(\sqrt{11}k\). For simplicity, we can take:
- \(\text{Opposite side} = 1\)
- \(\text{Hypotenuse} = \sqrt{11}\)

• Apply Pythagoras' theorem to calculate the Adjacent side:
\[ \text{Adjacent}^2 = \text{Hypotenuse}^2 - \text{Opposite}^2 \]
\[ \text{Adjacent}^2 = (\sqrt{11})^2 - (1)^2 \]
\[ \text{Adjacent}^2 = 11 - 1 = 10 \]
\[ \text{Adjacent} = \sqrt{10} \]

• Now, write down the formula for \(\cot \theta\):
\[ \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} \]

• Substitute the values we found:
\[ \cot \theta = \frac{\sqrt{10}}{1} = \sqrt{10} \]


Step 4: Final Answer:
The value of \(\cot \theta\) is \(\sqrt{10}\). This corresponds to option (C).