The simplest trigonometric equations

The arcsine of a ∈ [−1, 1] is the number in [−π/2, π/2] whose sine is a; the arccosine is the number in [0, π] whose cosine is a; the arctangent of any a is the number in (−π/2, π/2) whose tangent is a. For example arcsin(1/2) = π/6, arccos(−1/2) = 2π/3, arctan 1 = π/4. The equations: sin x = a (|a| ≤ 1) gives x = (−1)ᵏ arcsin a + πk, k ∈ ℤ; cos x = a (|a| ≤ 1) gives x = ± arccos a + 2πk, k ∈ ℤ; tan x = a gives x = arctan a + πk, k ∈ ℤ. If |a| > 1, sin x = a and cos x = a have no solutions since sine and cosine stay in [−1, 1]. Special cases: sin x = 0 gives x = πk, sin x = 1 gives x = π/2 + 2πk, sin x = −1 gives x = −π/2 + 2πk; cos x = 0 gives x = π/2 + πk, cos x = 1 gives x = 2πk, cos x = −1 gives x = π + 2πk. For a complex argument we substitute: for sin 3x = a take t = 3x, then recover x.