explain why the value of the sine ratio for an acute angle of a right triangle must always be a positive…

Answer:

The sine of an acute - angle in a right - triangle is the ratio of the length of the opposite side to the length of the hypotenuse. Since the opposite side is always shorter than the hypotenuse (by the Pythagorean theorem, (a^{2}+b^{2}=c^{2}), where (c) is the hypotenuse and (a,b) are the legs of the right - triangle, so (c > a) and (c > b)) and lengths are positive, the ratio (\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}) is positive (because both the opposite side and the hypotenuse have positive lengths) and less than 1 (because the numerator is less than the denominator).

Explanation:

Step1: Recall sine ratio definition

(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}})

Step2: Consider side - length properties

In a right - triangle, (\text{opposite}<\text{hypotenuse}) and (\text{opposite}>0,\text{hypotenuse}>0)

Step3: Analyze the ratio

Since (\text{opposite}>0,\text{hypotenuse}>0), the ratio is positive. Since (\text{opposite}<\text{hypotenuse}), the ratio is less than 1.