Question

If $ \tan \theta = \frac{3}{4} $, find the value of $ \sin \theta $.

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

Hint:

Remember: Use the Pythagorean theorem to find the hypotenuse when you are given the sides of a right triangle, and use this to calculate \( \sin \theta \).

Answer 1

The Correct Option is A

Step 1: Apply the tangent identity.
Given \( \tan \theta = \frac{3}{4} \). The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right triangle:\[\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}\]
Step 2: Employ the Pythagorean theorem.
To determine \( \sin \theta \), the hypotenuse is required. The Pythagorean theorem is utilized:\[\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2\]\[\text{hypotenuse}^2 = 3^2 + 4^2 = 9 + 16 = 25\]\[\text{hypotenuse} = \sqrt{25} = 5\]
Step 3: Compute \( \sin \theta \).
The value of \( \sin \theta \), the ratio of the opposite side to the hypotenuse, can now be calculated:\[\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}\]
Answer:
Consequently, \( \sin \theta = \frac{3}{5} \). The correct answer is option (1).