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.

The sine ratio is the length of the side opposite a given acute angle divided by the length of the hypotenuse. Because the hypotenuse is the side opposite the largest angle, the 90° angle, it has to be the longest side. Thus, the ratio will have a denominator that is larger than the numerator, and the ratio will be less than 1.

The sine of an angle x is defined as the ratio of the opposing side to the hypotenuse, in a right triangle having x as one of its acute angles. If it was greater than 1, it would mean the opposing side was longer than the hypotenuse. Try to draw a right triangle with one of the sides longer than the diagonal. You'll notice it's impossible. So the sine cannot be greater than 1.

Fitting the triangle into a circle of radius 1, such that the angle x is located at the origin and the hypotenuse is a radius of the circle, you can define "sine of x" for any angle. Since the triangle may end up flipped in any direction, including the negative x and y axis, it turns out that the sine of any number is between - 1 and + 1.