Question

If \( \tan \theta = \frac{3}{4} \) and \( \theta \) is acute, find \( \sin 2\theta \):

• A. \( \frac{24}{25} \)
• B. \( \frac{7}{25} \)
• C. \( \frac{3}{5} \)
• D. \( \frac{8}{25} \)

Hint:

To find \( \sin 2\theta \), remember the identity \( \sin 2\theta = 2\sin \theta \cos \theta \). Construct a right triangle when a trigonometric ratio is given to find other ratios.

Answer 1

The Correct Option is A

Step 1: Use identity for \( \sin 2\theta \).
\[ \sin 2\theta = 2 \sin \theta \cos \theta \] Step 2: Use triangle to find \( \sin \theta \) and \( \cos \theta \).
Given: \( \tan \theta = \frac{3}{4} \Rightarrow \frac{\text{opposite}}{\text{adjacent}} \).
Let opposite = 3, adjacent = 4. Then hypotenuse: \[ \text{hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] \[ \sin \theta = \frac{3}{5}, \quad \cos \theta = \frac{4}{5} \] Step 3: Plug values into identity. \[ \sin 2\theta = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25} \]