Inverse Tangent

Pre-Calculus · Axiom Academy

LESSON The Inverse Tangent Function Tangent hits every output infinitely often — so we keep just one branch, flip it across the line y = x , and read angles back out of slopes. 1. Tangent Repeats — So It Has No Inverse Yet The tangent graph is a chain of identical branches, one every units. Pick any output value and draw a horizontal line at that height: it crosses branch after branch, forever. Each crossing is a different angle with the same tangent, so "the angle whose tangent is 1 " has infinitely many answers — and a function is only allowed one. One output value, infinitely many input angles We keep the single branch through the origin, the open interval , and throw the rest away. Watch the point climb that branch: tangent rises the whole way (its slope, , is always positive), so every height is hit exactly once . And as x nears either edge, the curve shoots toward — meaning this one branch already produces every real output. are excluded — tangent is undefined there. Vertical asymptotes, never touched. Always climbing each output occurs once one-to-one (passes the horizontal-line test). Output runs from up to on this branch alone, so no value is missed. One-to-one and onto all reals — the exact conditions an inverse needs. On this branch, " " has the single answer — the solution lives on a discarded branch. One branch, one answer.

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