Question

analyzing the sine ratio
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.

Explanation:

1. First, recall the definition of the sine ratio for an acute angle $\theta$ in a right triangle: $\sin(\theta) = \frac{\text{opposite side length}}{\text{hypotenuse length}}$.
2. In a right triangle, all side lengths are positive values, so the ratio of two positive numbers is positive.
3. By the Pythagorean theorem, the hypotenuse is always the longest side of a right triangle. This means the length of the opposite side (a shorter leg) is always less than the hypotenuse length.
4. A positive fraction where the numerator is smaller than the denominator will always have a value less than 1.

Answer:

For an acute angle in a right triangle, the sine ratio is defined as $\frac{\text{opposite side}}{\text{hypotenuse}}$. All side lengths are positive, so the ratio is positive. The hypotenuse is the longest side of the triangle, so the opposite side (a shorter leg) is always shorter than the hypotenuse. A positive fraction with a smaller numerator than denominator is always a positive value less than 1.