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: Use the identity for tangent
We are given that \( \tan \theta = \frac{3}{4} \). By definition, the tangent of an angle is 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: Use the Pythagorean theorem
To find \( \sin \theta \), we need to find the hypotenuse. We can use the Pythagorean theorem: \[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \] \[ \text{hypotenuse}^2 = 3^2 + 4^2 = 9 + 16 = 25 \] \[ \text{hypotenuse} = \sqrt{25} = 5 \]
Step 3: Calculate \( \sin \theta \)
Now, we can calculate \( \sin \theta \), which is the ratio of the opposite side to the hypotenuse: \[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} \]
Answer:
Therefore, \( \sin \theta = \frac{3}{5} \). So, the correct answer is option (1).