What geometric principal allows us to define trigonometric functions as ratios of sides of triangles and 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?

Solution:

The  geometric principal which allows us to define trigonometric functions as ratios of sides of triangles and they are indeed functions

1. Similarity of Two Right Triangles

2. Concept of unit circle

→→As a complete circle means angle of π radians or an angle of 360°.So if you take any of trigonometric function and any right angled triangle , you will find that for angle between 0°≤Ф≤90°, values of trigonometric functions are different.Because for each angle, ratio of side lengths of right angled triangle changes, that's why for each unique angle we get a unique output value for different trigonometric functions.

You can call it  one-one onto mapping (function) also.