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how can the radian measure of an angle determine the arc length on the unit circle?
how can the unit circle be used to apply trigonometric functions to all real numbers?

Step-by-step explanation:

The definition of one radian is "a unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius." So, on the unit circle, because the radius is one, one radian's arc length is also one. Basically, on the unit circle, the radian measure of an angle is equal to the arc length is makes.

Trig Functions are just the coordinates on the unit circle plotted on to a graph. Think about it this way, if you pin a point on the unit circle and spin the circle, the point will go up and down and up and down forever (if there is no friction). This is the same thing with trig functions. They repeat them selves over and over again. In an actual triangle, we can't get angles more than 180 degrees, but because the trig functions go on forever, we can virtually plug in any angle degrees we want and produce an answer. The unit circle help us understand that some angles are just other angles written in another way, like 90 degrees and 450 degrees are actually the same thing.

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step-by-step explanatio89