In the Cartesian coordinate system, an equation of the form defines a unit circle with center .
An unit circle centered at the origin can be used to calculate values for the basic trigonometric functions. Suppose we draw a ray starting from the origin and meeting the positive x-axis with an angle of . If we drop a perpendicular from the point of intersection between the ray and the circle, we have a right triangle with hypotenuse of .
Using the definitions and , we find that and .
We can read off values for sine and cosine of an angle this way; we can draw the angle and approximate the x and y coordinates of the intersection.
We can also prove one of he fundamental theorems of trigonometry: . The proof is as follows:
We see that the length and width of the triangle in the diagram at left are and , respectively. We use the pythagorean theorem to get:
In polar form, the solution to this polynomial can be expressed as , where . This is commonly written as . Additionally, the solution points form a regular -gon on the unit circle.
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