Question

If \( \tan \theta = \frac{4}{3} \), find the value of \( \sin \theta \) in the first quadrant.

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

Hint:

For trigonometric ratios, construct a right triangle using the given ratio, calculate the hypotenuse, and use quadrant rules to determine signs.

Answer 1

The Correct Option is B

Given \( \tan \theta = \frac{4}{3} \) in the first quadrant, where all trigonometric functions are positive. Let the opposite side be 4 and the adjacent side be 3 in a right triangle. The hypotenuse is: \[ \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \] \[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{4}{5} \] Verify: \[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{3}{5}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \] Thus, the value of \( \sin \theta \) is: \[ \boxed{\frac{4}{5}} \]