Inverse trigonometry answers the question: given a ratio, what angle produced it? Where sin(30°) = 0.5, its inverse arcsin(0.5) = 30°. This calculator covers all six circular inverse functions — arcsin, arccos, arctan, arccot, arcsec and arccsc — plus the three hyperbolic inverses (arcsinh, arccosh, arctanh). Every result comes with the principal-value angle in both degrees and radians, a full step-by-step derivation, the general solution set (all co-terminal angles), and a unit-circle reference table.
How it works
Each inverse function is the left-inverse of the corresponding forward function, restricted to a range that makes the output unique:
| Function | Formula used | Principal range |
|---|---|---|
| arcsin(x) | built-in asin | [−90°, 90°] |
| arccos(x) | built-in acos | [0°, 180°] |
| arctan(x) | built-in atan | (−90°, 90°) |
| arccot(x) | π/2 − arctan(x) | (0°, 180°) |
| arcsec(x) | arccos(1/x) | [0°, 90°) ∪ (90°, 180°] |
| arccsc(x) | arcsin(1/x) | [−90°, 0) ∪ (0°, 90°] |
| arcsinh(x) | ln(x + √(x² + 1)) | (−∞, ∞) |
| arccosh(x) | ln(x + √(x² − 1)) | [0, ∞) |
| arctanh(x) | ½ · ln((1 + x)/(1 − x)) | (−∞, ∞) |
After computing the principal value the calculator derives the general solution — the full infinite family of angles that satisfy the equation — and presents it using the standard ± 360°·n or ± 180°·n formulas depending on the function’s period.
For the hyperbolic inverses the logarithm forms are evaluated directly, producing dimensionless real numbers rather than degree angles. The derivation panel shows every intermediate step so you can follow and verify the arithmetic by hand.
Worked example
Find the angle whose sine is 0.5:
- We need θ such that sin(θ) = 0.5.
- arcsin(0.5) = 30° = π/6 rad (principal value, in [−90°, 90°]).
- Verify: sin(30°) = 0.5 ✓
- General solution: θ = 30° + 360°·n or θ = 150° + 360°·n for any integer n.
Find the angle whose tangent is √3 ≈ 1.73205:
- arctan(1.73205) = 60° = π/3 rad.
- Verify: tan(60°) = √3 ≈ 1.73205 ✓
- General solution: θ = 60° + 180°·n (tangent has period 180°).
Hyperbolic example — arcsinh(1):
- Formula: arcsinh(1) = ln(1 + √(1² + 1)) = ln(1 + √2) = ln(2.41421…) ≈ 0.88137.
- Verify: sinh(0.88137) = (e^0.88137 − e^(−0.88137)) / 2 ≈ 1.0000 ✓
Formula note
For the reciprocal inverses the identities arcsec(x) = arccos(1/x) and arccsc(x) = arcsin(1/x) follow directly from the definitions sec = 1/cos and csc = 1/sin. For arccot the identity arccot(x) = π/2 − arctan(x) holds for all real x (using the principal-value convention where arccot has range (0°, 180°)). The hyperbolic inverse formulas are derived from the exponential definitions of sinh, cosh and tanh by solving for the exponent — the logarithm expressions are algebraically exact.
All arithmetic runs entirely in your browser using IEEE 754 double-precision floating point — no data is sent anywhere.