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. Definition of Sine Ratio: For an acute angle $\theta$ in a right triangle, $\sin\theta = \frac{\text{opposite side}}{\text{hypotenuse}}$.
2. Positivity: In a right triangle, all side lengths are positive real numbers. The ratio of two positive numbers is positive.
3. Less Than 1: The hypotenuse is the longest side of a right triangle, so the length of the opposite side (a leg) is always shorter than the hypotenuse. A positive fraction where the numerator is smaller than the denominator has a value less than 1.

Answer:

For any 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 numerator (opposite side) is always smaller than the denominator (hypotenuse), making the ratio a positive value less than 1.