Why is sin⁻¹(1/2) just π/6 and not the infinitely many angles that also have sine 1/2?
Sine is many-one — countless angles share the same sine — so a plain 'inverse' wouldn't be a function (one input, many outputs).
Sine is many-one — countless angles share the same sine — so a plain 'inverse' wouldn't be a function (one input, many outputs). To fix this, we restrict sin⁻¹ to a principal branch, the range [−π/2, π/2], where sine is one-one. Within that branch only one angle, π/6, has sine 1/2, so that's the principal value we report. The other angles still have sine 1/2, but they lie outside the agreed principal range.
Key point
Inverse trig uses a principal branch (sin⁻¹ ∈ [−π/2, π/2]) so it returns a single value.
Common mistake
listing every angle instead of the single principal value.
Memory tip
stick to the principal range and give the one angle inside it.
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