tan⁻¹ and cot⁻¹

Tangent is defined for x ≠ (2n + 1)π/2 and takes every real value. On the open interval (−π/2, π/2) it climbs from −∞ to ∞ exactly once, so tan⁻¹ : R → (−π/2, π/2).

The endpoints are left out because tan is undefined at ±π/2. The graph of tan⁻¹ x approaches the lines y = π/2 and y = −π/2 but never reaches them.

tan⁻¹ is odd and increasing: tan⁻¹(−1) = −π/4, tan⁻¹ 0 = 0, tan⁻¹ √3 = π/3.

Cotangent is defined for x ≠ nπ and takes every real value. On (0, π) it falls from ∞ to −∞ exactly once, so cot⁻¹ : R → (0, π).

Like cos⁻¹, cot⁻¹ never returns a negative angle. cot⁻¹(−1) = 3π/4, and cot⁻¹(−x) = π − cot⁻¹ x.

Both tan⁻¹ and cot⁻¹ accept every real number, unlike sin⁻¹ and cos⁻¹, which only accept [−1, 1].