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.

Answer 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.

Answer 2

Sine ratio for an acute angle of a right triangle always be a positive value less than 1.

In a right angled Δ ABC, sine of angle  $\theta$ can be written as:

$\rm Sin \; \theta = Perpendicular /Hypotenuse$

Therefore,

$\rm Sin \; \theta =AB/AC$  

Please refer to the diagram that is uploaded in the solution.

The Pythagoras theorem can be used to determine the unknown value in a right triangle from two known values.

Thus, the formulated way to represent the theorem for hypotenuse is given below:    

$\rm H = \sqrt{( P^2 + B^2) } .....(1)$

Hence,

Hypotenuse is the longest side of the Right  triangle and therefore its value must be greater than zero and should always be more than perpendicular.

Thus,

$\rm Hypotenuse > 0 \; and\; Hypotenuse > Perpendicular$

$\rm Sin \theta = \dfrac{Perpendicular }{Hypotenuse} <1$

so, we can say that $\rm Sin\theta$ is always positive and   less then  1.

For more information please refer to the link below

https://brainly.com/question/15141309