Use arcsin, arccos, and arctan to find angles from trig ratios.
The inverse trig functions arcsin (sin⁻¹), arccos (cos⁻¹), and arctan (tan⁻¹) answer the question: "which angle has this trig value?" Because trig functions are periodic, we restrict their domains to make them invertible.
Domain and Range of Inverse Trig
Each function is restricted to give exactly one output per input.
arcsin(√2/2): which angle in [−π/2, π/2] has sine = √2/2? Answer: π/4 (45°). arccos(−1/2): which angle in [0, π] has cosine = −1/2? Answer: 2π/3 (120°).
arcsin(sin θ) = θ ONLY if θ is in [−π/2, π/2]. For angles outside this range, the result is the reference angle equivalent in the principal range.
Remember This!
On a calculator: sin⁻¹, cos⁻¹, tan⁻¹ buttons give the principal value. For other quadrant solutions, use reference angle symmetry: if sin θ = k, the second solution is π − arcsin(k).
What is arctan(1) in radians?