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 length of the leg opposite an angle to the length of the hypotenuse. since the hypotenuse is the side opposite the largest angle, it must be the longest side. thus, the ratio will have a

Explanation:

Step1: Recall sine - definition

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, where $\theta$ is an acute angle. The lengths of the sides of a triangle are positive, so the sine ratio is positive.

Step2: Analyze hypotenuse length

The hypotenuse is the side opposite the right - angle (the largest angle in a right - triangle). By the triangle inequality theorem, the hypotenuse is the longest side of a right - triangle. Let the length of the side opposite the acute angle be $a$ and the length of the hypotenuse be $c$. Then $a>0$ and $c > a$ (since $c$ is the hypotenuse). So, $\sin\theta=\frac{a}{c}$ and $0 <\frac{a}{c}<1$.

Answer:

In a right triangle, sine is the ratio of the length of the leg opposite an angle to the length of the hypotenuse. Since the hypotenuse is the side opposite the largest angle, it must be the longest side. Thus, the ratio will have a positive value less than 1.