What are inverse trig functions?
arcsin, arccos and arctan run the trig functions backwards: given a ratio they return the angle that produces it. For example arcsin(0.5) = 30° because sin 30° = 0.5.
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Inverse Trigonometric Functions
Evaluate arcsin, arccos and arctan with their principal-value ranges, returning the angle in degrees or radians, with the full reasoning shown.
arcsin x ∈ [−90°,90°]; arccos x ∈ [0°,180°]; arctan x ∈ (−90°,90°)
Angle unit
Frequently asked questions
Why do they have restricted ranges?
Because sin, cos and tan repeat, an inverse would otherwise have infinitely many answers. Each is given one principal range: arcsin and arctan return [−90°, 90°], arccos returns [0°, 180°].
Can you show a worked example?
arccos(0.5): the angle in [0°,180°] whose cosine is 0.5 is 60°. In radians that is π/3 ≈ 1.0472 rad; switch the toggle and the answer is shown in radians.
What inputs are allowed?
arcsin and arccos only accept values from −1 to 1, since sine and cosine never exceed that range. arctan accepts any real number.
Where are they used?
Finding an angle from a ratio in navigation, working out a launch or view angle, and defining the principal branches used throughout calculus.