Which trigonometric function would you use to find the opposite side in a right triangle given the angle and hypotenuse?

In a right triangle, when you want to find the length of the opposite side relative to a given angle and you know the length of the hypotenuse, the sine function is the appropriate choice to use. The sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

Mathematically, this relationship can be expressed as:

[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} ]

From this equation, you can rearrange it to find the opposite side:

[ \text{opposite} = \sin(\theta) \times \text{hypotenuse} ]

This formula directly shows how to calculate the length of the opposite side when the angle and hypotenuse are known.

The other trigonometric functions serve different purposes: cosine relates the adjacent side to the hypotenuse, tangent relates the opposite side to the adjacent side, and arc functions are generally used for finding angles from given ratios. Thus, in the context of this question, the sine function is clearly the correct choice for determining the length of the opposite side when given the angle and hypotenuse.