Question

complete the statements to explain why the value of the sine ratio for an acute angle of a right triangle must always be a positive value less than 1. in a right triangle, sine is the ratio of the dropdown options: length of the leg opposite an angle to the length of the hypotenuse, length of the leg adjacent to an angle to the length of the hypotenuse, length of the leg opposite an angle to the length of the leg that is adjacent to the angle since the hypotenuse is the side opposite the large

Explanation:

Step1: Recall sine - ratio definition

In a right - triangle, the sine of an acute angle is defined as the ratio of the length of the leg opposite the angle to the length of the hypotenuse.

Step2: Analyze side - length relationships

In a right - triangle, the hypotenuse is the longest side. Let the length of the leg opposite the acute angle be $a$, and the length of the hypotenuse be $c$. Then $\sin\theta=\frac{a}{c}$, where $\theta$ is the acute angle. Since $a\gt0$ (length is non - negative and in a non - degenerate triangle, the side has positive length) and $c > a$ (hypotenuse is the longest side), $0<\frac{a}{c}<1$.

Answer:

length of the leg opposite an angle to the length of the hypotenuse