The unit circle is a circle of radius one placed at the origin of the coordinate system. The properties of a point formed by an angle on the unit circle correspond to the input and output of the trigonometric functions.
Each of the functions takes in an angle measured in radians as input and returns a ratio as output. The significance of the unit circle and its radius of length one is that output of the functions corresponds directly with the geometry of the point formed by the angle. These lengths are labeled in the figure below.
In the case of an angle formed on a general circle, the same trigonometry can be observed by adding the variable to represent the arc length formed by the angle and the variable to represent the radius of the circle.
In this case, we know that an angle measured in radians is equal to the arc length divided by the length of the radius of the circle[1].
Similarly, the output of sine is equal to the ratio of the vertical component of the point divided by the radius and the output of cosine is equal to the horizontal component of the point divided by the radius.
Substituting into the general case and making the circle a unit circle we can observe two things happen. First, the arc length is the same value as the angle measured in radians. Second, the output of the functions, which can be thought of as a ratio, now corresponds directly to the coordinates of the point formed by the angle.
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Radian Angle System Concept