The Tangent Graph

Tangent is defined as tan(theta) = sin(theta)/cos(theta). Sine and cosine are smooth, continuous waves, but dividing by cosine causes trouble whenever cosine equals zero. That happens at theta = π/2, 3pi/2, and every odd multiple of π/2 going in either direction. At those angles the ratio blows up, so the graph has a vertical asymptote there instead of a defined value. Between consecutive asymptotes the function climbs steadily from negative infinity to positive infinity, crossing zero at the midpoint where sine itself is zero. That midpoint is an inflection point, where the graph passes through the x-axis and switches concavity.

Graphing y = tan(x) from -π to π: asymptotes sit at x = -π/2 and x = π/2, and the curve crosses the x-axis with an inflection point at x = 0. Each branch between asymptotes looks the same, just shifted sideways, because the pattern of sine and cosine signs repeats every half turn instead of every full turn. That is the key to the period: rotating an angle by π flips the sign of both sine and cosine, but a ratio of two negatives is positive again, so tan(theta + π) = tan(theta) exactly. Sine and cosine need a full 2pi turn to return to their starting values, but tangent only needs π.

This sets tangent apart from sine and cosine, which stay continuous and bounded between -1 and 1. Tangent is unbounded and discontinuous, but it still connects back to the unit circle: tan(theta) is literally the slope of the terminal ray, and that slope repeats every half rotation, which is exactly why the graph's period is π.