Trigonify

Topics · 3 identities

Inverse

Domains and ranges of arcsine, arccosine, and arctangent.

arcsin range and domain: [−1, 1] → [−π/2, π/2]

arcsin⁡:[−1,1]→[−π2,π2]\arcsin : [-1, 1] \to [-\tfrac{\pi}{2}, \tfrac{\pi}{2}]

arcsin takes inputs in [−1, 1] and returns angles in [−π/2, π/2]. Sine repeats, so it has no true inverse until the range is restricted this way - which is why a calculator's arcsin never returns an obtuse angle, and why the Law of Sines' ambiguous case needs checking by hand.

Open identity

arccos range and domain: [−1, 1] → [0, π]

arccos⁡:[−1,1]→[0,π]\arccos : [-1, 1] \to [0, \pi]

arccos takes inputs in [−1, 1] and returns angles in [0, π]. That range covers every angle a triangle can have, which is why the Law of Cosines never suffers an ambiguous case while the Law of Sines does.

Open identity

arctan range and domain: ℝ → (−π/2, π/2)

arctan⁡:R→(−π2,π2)\arctan : \mathbb{R} \to (-\tfrac{\pi}{2}, \tfrac{\pi}{2})

arctan accepts every real number and returns an angle in (−π/2, π/2). The interval is open, not closed: tangent has asymptotes at ±π/2, so those endpoints are approached but never reached.

Open identity