The Tangent Graph

tan(x) = sin(x)/cos(x), the height of the point on the unit circle divided by its x. It is 0 wherever the sine is 0: at 0, π, 2π and every integer multiple of π. Where the cosine is 0, at π/2 and 3π/2, it has no value at all: you cannot divide by 0.

Between those gaps the graph climbs. Just right of −π/2 the cosine is a tiny positive number, so the tangent is huge and negative; it rises through 0 at 0, through 1 at π/4, and shoots up towards π/2. Then the next branch starts over from far below. Each branch is a copy of the last, so the tangent repeats every π, not every 2π.

The sign follows the circle: positive where sine and cosine agree (quadrants I and III), negative where they differ (quadrants II and IV).