Question

What geometric principal allows us to define trigonometric functions as ratios of sides of triangles and to be confident that they are indeed functions? That is, how do we know that the value of each angle put into a trigonometric function results in exactly one output value

Answer 1

@ganeshie8 @Michele_Laino @freckles What in the blazes are they asking here? 0_o

Answer 2

isnt it just the pythagorean theorum

Answer 3

I think because the angle formed is a function of the radius, the radius is the length

Answer 4

I think the geometric principal that allows us to be confident that "each angle produces an unique trig output" is the concept of "similar triangles".

Answer 5

For example, consider all the similar triangles with angles : \(40,50,90\)
Since the triangles are similar, the corresponding sides form a proportion. So no matter what the scale factor is, the trig ratio for a particular angle is always same.

Answer 6

agree with @ganeshie8 you solved my "missing piece" the trig ratio is a comparison of the lengths and the similar figures will "equate" the different lengths so the trig ratio for a particular angle will always be the same

Answer 7

I think Pythagoras theorem allows us to use similar figures in the first place.