Graphing Tangent and Other Trig Functions

Calculus Readiness · Axiom Academy

Why tangent isn't a wave — it's a ratio that shoots to infinity, with asymptotes wherever cosine hits zero, repeating every . 1. Tangent Is a Ratio That Blows Up Take a point on the unit circle at angle θ . Its height is and its base is . Tangent is their ratio . Watch what happens as θ sweeps toward : the base shrinks toward zero, so the ratio shoots upward without bound — straight into a vertical wall. height ÷ base — the numerator is sin θ 2. The Parent Graph, Branch by Branch Stack those ratios at every angle and the parent graph appears as a chain of identical branches . Each branch lives between two asymptotes, rises from up through a zero to , then the next branch starts over from the bottom. The tracer climbs one branch, vanishes at the wall, and reappears from on the next. At , where . Spaced exactly apart. At — the midpoint of each branch, where . Every branch climbs left to right; tangent never turns around inside a branch. Range is all of (unbounded). Odd: . Five-point skeleton per branch On the branch between and : , , . Plot those three points, draw the two asymptotes, and the S-shaped branch draws itself. Here is the surprise: tangent repeats twice as fast as sine or cosine. Take any single branch and slide it sideways by exactly — it lands perfectly on the next branch. The reason is the ratio: rotating flips both and , and the two sign flips cancel. The mechanism: and , so the two negatives cancel in the ratio.

This is the written version of the interactive lesson above. See the full Calculus Readiness course.