Inverse Trigonometric Functions

Calculus Readiness · Axiom Academy

LESSON Inverse Trigonometric Functions Going from a ratio back to an angle — and why sine, cosine, and tangent must be tamed onto one branch before they can be undone. 1. Why We Have to Restrict the Domain A function can be undone only if it is one-to-one — each output comes from at most one input. But , , and repeat every (or ), so they are about as far from one-to-one as a function gets. Watch the line sweep across the sine wave: it strikes the curve again and again. Every hit is an angle whose sine is — so " " can't choose among them. Infinitely many angles share the ratio The cure: keep only the rising branch where sine is one-to-one 2. Reflect the Branch to Get Arcsine Reflecting a graph across the line y = x swaps every point's coordinates — exactly what an inverse does. Fold the kept branch of over that mirror and you get : a curve that takes a ratio on the horizontal axis and returns the angle on the vertical one. To evaluate , go in at , rise to the curve, and read off the angle. The same reflection idea, applied to each parent (on its own one-to-one branch), gives: — the angle whose sine is x . — the angle whose cosine is x . — the angle whose tangent is x . Each inverse lives inside a fixed band of output angles — chosen so every legal input gets exactly one answer, while keeping as much of the parent curve as possible. Watch the three ranges paint in: rides , rides , and flattens toward the open asymptotes at .

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