Yes, the tangent function is periodic.
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The tangent function is a periodic function with period 180 degrees sotan(360) = tan(360-2*180) = tan(0) = 0.
No.
When you graph a tangent function, the asymptotes represent x values 90 and 270.
Because the tangent is a function of with the angle as its argument.
Tangent = sine/cosine provided that cosine is non-zero. When cosine is 0, then tangent is undefined.