How do you know that the value of each angle put into a trigonometric function results in exactly one output value?

The value of each angle put into a trigonometric function results in exactly one output value, because that angle represents a single set of x and y coordinates on the ray at the end of the unit circle. Since the trigonometric functions are all defined as the ratio of x and/or y and/or 1, there can only be one output value for each angle.

However, the reverse is not true. As an example, tangent is defined as sine over cosine, or y over x. This means that an angle of theta plus 180 degrees generates the same value, because y over x is the same as -y over -x.