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Look at the attached figure. We define the unit circle as a circle with center in the origin (0,0) and radius 1.
Then, we consider a point P on the circumference. We call the angle between the positive half of the x axis and the radius AP.
We define
As you can see, ACD is a right triangle, and so we have
But since we know that AD is the cosine, AC is the sine, and AP is the radius (which is 1, and remains 1 when squared), we have just found out that
