Question

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:

In a right - triangle, for an acute angle \(A\), \(\sin A=\frac{\text{opposite}}{\text{hypotenuse}}\). The opposite side and hypotenuse are lengths of sides of the triangle. Lengths are positive, so \(\sin A>0\). Also, by the triangle inequality, the length of the opposite side is less than the length of the hypotenuse (\(\text{opposite}<\text{hypotenuse}\)), so \(\frac{\text{opposite}}{\text{hypotenuse}}< 1\).

Answer:

The value of the sine ratio for an acute angle of a right - triangle is positive because it is a ratio of two positive lengths (opposite side and hypotenuse). It is less than 1 because the length of the side opposite an acute angle in a right - triangle is always less than the length of the hypotenuse.