Question

If \( \tan \theta = \frac{3}{4} \), then \( \sin \theta \) is

• A. \(\frac{4}{5} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{4}{3} \)
• D. \( \frac{3}{5} \)

Hint:

To find trigonometric ratios, use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) and solve for the unknown ratio.

Answer 1

The Correct Option is D

Step 1: Given \( \tan \theta = \frac{3}{4} \), we can use the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Step 2: Let \( \sin \theta = 3k \) and \( \cos \theta = 4k \) for some constant \( k \). Step 3: Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \): \[ (3k)^2 + (4k)^2 = 1 \] \[ 9k^2 + 16k^2 = 1 \] \[ 25k^2 = 1 \] \[ k^2 = \frac{1}{25} \] \[ k = \frac{1}{5} \] Step 4: Therefore, \( \sin \theta = 3k = \frac{3}{5} \).
Thus, the correct answer is \( \boxed{\frac{3}{5}} \).