A 30° kick has a hypotenuse of 10". Given that the cosecant of 30° is 2, determine the side opposite, or how far off the surface the end of the kick needs to be?

To determine the height of the side opposite the angle in a right triangle relative to a 30° angle where the hypotenuse is 10 inches, we can use the relationship defined by the cosecant function. The cosecant of an angle in a right triangle is defined as the ratio of the length of the hypotenuse to the length of the side opposite the angle.

Given that the angle is 30° and the cosecant of 30° is 2, we can set up the equation as follows:

[

\csc(30°) = \frac{\text{hypotenuse}}{\text{opposite side}}

]

Substituting in the known values:

[

2 = \frac{10}{\text{opposite side}}

]

To isolate the opposite side, we can rearrange the equation:

[

\text{opposite side} = \frac{10}{2} = 5 \text{ inches}

]

This means the height from the surface to the end of the kick, which represents the length of the side opposite the 30° angle, needs to be 5 inches. This aligns perfectly with the correct answer. Thus, the side opposite is indeed