In a right triangle, which of the following expressions represents $\sin \theta$ for an acute angle $\theta$?

• A. $\frac{\text{hypotenuse}}{\text{opposite}}$
• B. $\frac{\text{adjacent}}{\text{hypotenuse}}$
• C. $\frac{\text{opposite}}{\text{adjacent}}$
• D. $\frac{\text{opposite}}{\text{hypotenuse}}$

Answer: (D)

Verified step by step guidance

1. Recall the definition of the sine function for an acute angle \( \theta \) in a right triangle: \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \).
2. Identify the sides of the triangle relative to the angle \( \theta \): the side opposite to \( \theta \), the side adjacent to \( \theta \), and the hypotenuse (the longest side opposite the right angle).
3. Compare each given expression to the sine definition: check if the numerator corresponds to the opposite side and the denominator corresponds to the hypotenuse.
4. Eliminate the options that do not match the sine ratio, such as \( \frac{\text{hypotenuse}}{\text{opposite}} \), \( \frac{\text{adjacent}}{\text{hypotenuse}} \), and \( \frac{\text{opposite}}{\text{adjacent}} \).
5. Conclude that the correct expression for \( \sin(\theta) \) is \( \frac{\text{opposite}}{\text{hypotenuse}} \), as it aligns with the fundamental trigonometric definition.