Here is a useful image of the unit circle labeled with the “special angles” and the coordinates of the corresponding points on the unit circle:
(via http://etc.usf.edu/clipart/43200/43215/unit-circle7_43215.htm)
This image is useful since you can use it to findĀ the sine and cosine of any of the given angles, usingĀ the definitions of sin t and cosĀ tĀ as the y- and x-coordinates, respectively, of the point on the unit circle corresponding to the angle t:
Below is a gif which may help you visualize the graphs of the sine and cosine functions in terms of their “unit circle definitions.”
First, here’s a reminder of the definitions of sine and cosine in terms of the unit circle: sin t is the y-coordinate of the corresponding point on the unit circle, and cos t is the x-coordinate (where t is the angle measured in radians).
Now look at the graphs of the coordinates as the point rotates around the circle:
(By LucasVB (Own work) [Public domain], via Wikimedia Commons)
Here’s what’s going on: if you look at the unit circle part, the blue dotĀ is at the x-coordinate of the point of the unit circle–and hence the blue graph is the graph of 
Similarly, the redĀ dotĀ is at the y-coordinate of the point of the unit circle, and hence the redĀ graph is the graph of 
(Watch one full “period”–from when the angleĀ 
For an interactive version, click through on the image below for a Desmos graph:
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