Inverse Trigonometric Calculator

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions reverse the operation of standard trigonometric functions. Where sin(θ) = x, the inverse function arcsin(x) = θ recovers the angle from a known ratio. This inverse trigonometric calculator evaluates all six principal inverse functions — arcsin, arccos, arctan, arccot, arcsec, and arccsc — and returns the angle in either degrees or radians.

The Six Inverse Trigonometric Functions

Each inverse function maps a ratio back to a unique angle within its principal value range, a restricted interval chosen to make the function one-to-one and therefore invertible. The six functions with their domains and ranges are:

• arcsin(x) — Domain: [−1, 1], Range: [−90°, 90°] or [−π/2, π/2]
• arccos(x) — Domain: [−1, 1], Range: [0°, 180°] or [0, π]
• arctan(x) — Domain: (−∞, +∞), Range: (−90°, 90°) or (−π/2, π/2)
• arccot(x) — Domain: (−∞, +∞), Range: (0°, 180°) or (0, π)
• arcsec(x) — Domain: |x| ≥ 1, Range: [0°, 90°) ∪ (90°, 180°]
• arccsc(x) — Domain: |x| ≥ 1, Range: [−90°, 0°) ∪ (0°, 90°]

Core Formula

The general notation for any inverse trigonometric operation is: θ = f⁻¹(x), where f ∈ {sin, cos, tan, cot, sec, csc}. Given a ratio x, the calculator finds the unique angle θ — within the principal value range — such that f(θ) = x. For example, arcsin(0.5) = 30° because sin(30°) = 0.5, and arctan(1) = 45° because tan(45°) = 1.

Domain Restrictions and Principal Values

Trigonometric functions are periodic, repeating the same ratio at infinitely many angles. To produce a single, unambiguous output, each inverse function restricts its output to a principal value range. According to Khan Academy's introduction to inverse trig functions, these principal value conventions are standardized across mathematics education worldwide. For arcsin and arccos, valid inputs must satisfy −1 ≤ x ≤ 1, since sine and cosine always output values within that interval. For arcsec and arccsc, valid inputs require |x| ≥ 1, because secant and cosecant — the reciprocals of cosine and sine — always have magnitudes at or above 1. The functions arctan and arccot accept any real number as input.

Identities and Relationships

The six inverse functions are connected through complementary identities. For all x in [−1, 1], the relationship arcsin(x) + arccos(x) = π/2 holds, meaning the two functions are complementary. Similarly, arctan(x) + arccot(x) = π/2 for all real x. Paul's Online Math Notes (Section 2.5: Inverse Trig Functions) provides rigorous proofs of these identities and their applications in calculus, including the derivatives of all six inverse functions.

Worked Examples

Apply the calculator to these representative inputs:

• arcsin(0.707) ≈ 45° — A sine ratio of approximately 0.707 (√2 ÷ 2) corresponds to a 45° angle in a right isoceles triangle.
• arccos(0.5) = 60° — When the cosine ratio equals 0.5, the angle is exactly 60°.
• arctan(√3) ≈ 60° — A tangent ratio of √3 ≈ 1.732 corresponds to a 60° angle.
• arcsec(2) = 60° — Because cos(60°) = 0.5, secant inverts to sec(60°) = 2, so arcsec(2) = 60°.

Real-World Applications

Inverse trigonometric functions appear across engineering, physics, navigation, and computer graphics:

• Structural engineering: Roof pitch angles are computed as arctan(rise ÷ run). A roof rising 4 feet over a 12-foot horizontal span has a pitch angle of arctan(4/12) ≈ 18.43°.
• Physics — Snell's Law: The refracted angle θ₂ is calculated as arcsin(n₁ × sin(θ₁) ÷ n₂), where n₁ and n₂ are the refractive indices of two media.
• Navigation: Bearing angles between GPS coordinates are derived using arctan(Δlongitude ÷ Δlatitude), adjusted for quadrant.
• Computer graphics: 3D rendering pipelines use arccos to compute the angle between a surface normal vector and a light direction vector for physically based shading.

Converting Between Degrees and Radians

The output unit can be set to degrees or radians depending on the application. To convert manually: Radians = Degrees × (π ÷ 180) and Degrees = Radians × (180 ÷ π). For instance, 60° equals 60 × (π/180) = π/3 ≈ 1.0472 radians. Radians are the standard unit in calculus and scientific computing, while degrees remain the conventional unit in applied engineering and everyday geometry.