Question

If \( 3 \cot \theta = 4 \), then the value of \( \cos \theta \) will be:

• A. \( \frac{4}{5} \)
• B. \( \frac{3}{5} \)
• C. \( \frac{5}{7} \)
• D. \( \frac{5}{4} \)

Hint:

For solving trigonometric equations involving cotangent, use the identity \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \).

Answer 1

The Correct Option is A

Step 1: Use the identity for cotangent.
We are given that: \[ 3 \cot \theta = 4 \quad \Rightarrow \quad \cot \theta = \frac{4}{3} \] Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), so: \[ \frac{\cos \theta}{\sin \theta} = \frac{4}{3} \]
Step 2: Use the identity \( \cos^2 \theta + \sin^2 \theta = 1 \).
Let \( \cos \theta = 4k \) and \( \sin \theta = 3k \), where \( k \) is a constant. From the identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we get: \[ (4k)^2 + (3k)^2 = 1 \] \[ 16k^2 + 9k^2 = 1 \] \[ 25k^2 = 1 \] \[ k^2 = \frac{1}{25} \quad \Rightarrow \quad k = \frac{1}{5} \]
Step 3: Calculate \( \cos \theta \).
Thus, \( \cos \theta = 4k = \frac{4}{5} \).
Step 4: Conclusion.
The value of \( \cos \theta \) is \( \frac{4}{5} \), so the correct answer is (A).