Arc Sine — sin⁻¹(x)
The inverse of sin restricted to [−π/2, π/2]. It "undoes" sine, returning the angle whose sine is x. The graph is an S-shaped curve through the origin, bounded above and below by the horizontal asymptotes y = ±π/2.
Key Properties
- sin⁻¹(sin x) = x for x ∈ [−π/2, π/2]
- sin(sin⁻¹ x) = x for x ∈ [−1, 1]
- sin⁻¹(−x) = −sin⁻¹(x) → Odd function
- Monotonically increasing on [−1, 1]
- Derivative: d/dx[sin⁻¹x] = 1/√(1−x²)
Arc Cosine — cos⁻¹(x)
Inverse of cosine restricted to [0, π]. A decreasing curve from (−1, π) to (1, 0), passing through (0, π/2). Note that sin⁻¹x + cos⁻¹x = π/2 for all x in [−1,1].
Key Properties
- sin⁻¹x + cos⁻¹x = π/2 for x ∈ [−1, 1]
- cos⁻¹(−x) = π − cos⁻¹(x)
- Monotonically decreasing on [−1, 1]
- Derivative: d/dx[cos⁻¹x] = −1/√(1−x²)
Arc Tangent — tan⁻¹(x)
Defined for all real x. The graph has two horizontal asymptotes at y = ±π/2, passes through the origin, and is S-shaped. It is one of the most important inverse trig functions in calculus and engineering.
Key Properties
- tan⁻¹(−x) = −tan⁻¹(x) → Odd function
- tan⁻¹x + cot⁻¹x = π/2
- Monotonically increasing on ℝ
- Derivative: d/dx[tan⁻¹x] = 1/(1+x²)
- As x→±∞, y→±π/2 (horizontal asymptotes)
Arc Cotangent — cot⁻¹(x)
Inverse of cotangent restricted to (0, π). Defined for all real x, it is a decreasing function. Closely related to arctan: cot⁻¹x = π/2 − tan⁻¹x for all x.
Key Properties
- tan⁻¹x + cot⁻¹x = π/2
- cot⁻¹(−x) = π − cot⁻¹(x)
- Monotonically decreasing on ℝ
- Derivative: d/dx[cot⁻¹x] = −1/(1+x²)
Arc Secant — sec⁻¹(x)
Inverse of secant. Defined only for |x| ≥ 1. The graph has two separate branches — one for x ≥ 1 (approaching π/2 from below) and one for x ≤ −1 (approaching π/2 from above). Note the gap in the domain at (−1, 1).
Key Properties
- sec⁻¹x + cosec⁻¹x = π/2 for |x| ≥ 1
- sec⁻¹(−x) = π − sec⁻¹(x)
- Defined only where |x| ≥ 1 — no value at x=0
- Derivative: d/dx[sec⁻¹x] = 1/(|x|√(x²−1))
Arc Cosecant — cosec⁻¹(x)
Inverse of cosecant. Like arcsec, it is defined only for |x| ≥ 1. The graph has two branches symmetric about the origin, and is an odd function. The range excludes 0.
Key Properties
- sec⁻¹x + cosec⁻¹x = π/2
- cosec⁻¹(−x) = −cosec⁻¹(x) → Odd
- Derivative: d/dx[cosec⁻¹x] = −1/(|x|√(x²−1))