Tangent as the Slope of the Ray

Picture the terminal ray of angle theta spinning out from the origin through the point (cos theta, sin theta) on the unit circle. Slope is rise over run, and here the rise is sin theta while the run is cos theta, measured from the origin. So tan theta = sin theta / cos theta isn't a separate fact to memorize — it's just the slope formula applied to this exact point. As theta approaches 90 degrees (π/2), cos theta shrinks toward 0 while sin theta stays near 1, so the ray gets closer and closer to vertical. A vertical line has no defined slope, which is why tan theta shoots off toward infinity and is undefined exactly at π/2.

Because tangent is a slope, and slope describes a whole line rather than a single direction, tangent treats opposite rays as identical. The ray at theta and the ray at theta + π point in opposite directions, but they sit on the same straight line through the origin, so they have the same rise-over-run and the same tangent value. Sine and cosine don't share this: they flip sign between theta and theta + π, since the point itself moves to the opposite side of the circle. That's the structural reason tangent repeats every π radians while sine and cosine need the full 2pi to return to their starting values.

This slope view sets up everything tangent does next: its graph as a stretched, repeating curve with vertical asymptotes wherever cos theta = 0, and its connection to the slopes of lines you already graph in algebra.