Inverse Cosine

Pre-Calculus · Axiom Academy

LESSON The Inverse Cosine Function Finding the angle from its cosine — why lives on , and why it runs downhill. Cosine sends an angle to a value: . Inverse cosine reverses the arrow — it answers "which angle has this cosine?" So . The animation feeds a value into the machine and reads back the angle on the restricted cosine branch. It returns the angle whose cosine is x For an inverse to exist, each value must come from exactly one angle. But on the sine restriction , cosine fails this badly: a vertical line at strikes the circle at two symmetric points — and — because . Two angles, one value: not invertible. Watch the value-line hit twice. . One input would need two outputs. On cosine rises to 1 at 0 , then falls — every value (except 1 ) is hit twice. Use only the top half of the circle, — there the value-line meets it once. sweeps the upper unit circle, where , so each x -value is reached a single time. Sine's branch is the right half of the circle (Q1, Q4); cosine needs the top half (Q1, Q2). Same idea, different slice — which is exactly why the two inverse ranges differ. Keep only the piece of over . It starts high at (0,1) and slides straight down to — strictly decreasing, never turning back. A horizontal value-line crosses this bright branch exactly once , so every value in [-1,1] has a single angle. That is the whole requirement for an inverse. Domain [-1,1] (the legal cosine values) maps to range (the angles, i.e. to ). 4. The Graph: Reflect Across y = x

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