Inverse Sine

Pre-Calculus · Axiom Academy

LESSON The Inverse Sine Function Sine sends an angle to a height. To run that backward — height to angle — we first have to make sine invertible by restricting where it lives. 1. Sine Fails the Horizontal Line Test To undo a function, each output must come from exactly one input. But run a horizontal line y = c across the sine wave: it strikes the curve again and again. A single height like is produced by infinitely many angles. one value, many angles — sine is not one-to-one 2. Restrict to One Rising Branch Keep only the arch from to — Quadrant IV up through Quadrant I. On this interval sine climbs steadily from -1 to 1 , never repeating a value. Every height is hit exactly once, so the branch is one-to-one and can be inverted. From to the curve only goes up — no two angles share a height. It sweeps the full output range [-1, 1] , so we lose no possible sine value. The allowed angles are exactly — the key band to watch. One height ↔ one angle. This is the branch will read backward. It's the simplest piece straddling 0 that climbs through every value once: symmetric about the origin and centered on the angles we use most. By convention this is the principal branch . An inverse is the original reflected over the line y=x — every point becomes . Flip the restricted arch and it sweeps into the graph of . The roles trade: sine's domain becomes arcsin's range, and sine's range [-1,1] becomes arcsin's domain. Domain: · Range: (that is, to )

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