Posted by Skye on Wednesday, December 19, 2012 at 5:08pm.
sin and cos change smoothly from 0 to 1, then 0 to -1 and back every 2pi
tan varies from 0 to infinity, flips over to minus infinity, and comes back to 0 after only pi.
tan(x+pi) = (tanx + tan pi)/(1-tanx*tanpi)
= (tanx + 0)/(1-0)
= tanx
so every pi tan comes back around. As the radius rotates around the 2pi unit circle, sin and cos make one period each. tan starts all over each time the slope is vertical, which happens twice during each 2pi period.
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