Module 2 · Lesson 2.3
Tangent & the Reciprocal Functions
Tangent isn't a separate mystery ratio — it's simply the slope of the radius as it spins around the unit circle.
Intuition
Draw the radius from the center of the unit circle out to the rotating point. That line has a slope — rise over run — just like any other line. Tangent is exactly that slope: how steep the radius is at a given angle.
Near 0°, the radius is nearly flat, so its slope is small. As the angle climbs toward 90°, the radius swings closer to vertical and its slope shoots toward infinity. This is exactly the calculation behind a line-of-sight or slope check in a physics engine — how steep is the line from one point to another?
Core concept
Since the point on the circle sits at $(\cos\theta, \sin\theta)$, the slope of the radius reaching it is rise over run: $\sin\theta$ over $\cos\theta$. That ratio is the tangent. Three more functions — cosecant, secant, and cotangent — are simply the reciprocals of sine, cosine, and tangent, useful whenever a calculation is easier upside down.
Interactive Lab
Drag the slider to spin the radius. The amber segment is the classic "tangent" line — its length is the tangent value, and it grows fast as the angle steepens.
Key Idea
Tangent is the slope of the spinning radius: rise ($\sin\theta$) over run ($\cos\theta$). The other three functions are simply reciprocals — cotangent flips tangent, secant flips cosine, and cosecant flips sine. Each becomes undefined exactly where its denominator hits zero.