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I realize that this question is incredibly basic for this website, but I really need help.

I took this image from

unit circle

It's a picture of the Unit Circle. On the outside, in purple, are Cartesian coordinates, and on the inside, in black, are degrees.

What process is taken to go from the degrees to the coordinates? What is the generalized process/algorithm that would be performed?

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See here for a geometrical way to find sin 18 degrees [missing from your picture] – user16697 Jan 2 '12 at 9:11
up vote 1 down vote accepted

Let's say you have an angle $\theta$ corresponding to a point on your circle, and you want to find the coordinates of that point on the circle. The coordinates are $(\cos \theta, \sin \theta)$ where $\sin$ is the sine function and $\cos$ is the cosine function. These can be calculated using most modern calculators, although usually one memorizes the values of these functions for "nice" angles like $0^\circ$ and $45^\circ$, i.e. the angles you most often encounter. You can read more about these functions on wikipedia.

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I think you mean $(\cos\theta,\sin\theta)$ - or $e^{i\theta} = \cos\theta + i\sin\theta$. – kahen Jan 2 '12 at 7:52
@kahen Yes, I missed that typo. Thanks. – Alex Becker Jan 2 '12 at 7:54

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