Inverse Trigonometric Functions Calculator
Principal values of sin⁻¹, cos⁻¹, tan⁻¹, cot⁻¹, sec⁻¹ and cosec⁻¹ as exact multiples of π, inverses of functions like sin⁻¹(sin 5π/6), and the tan⁻¹x ± tan⁻¹y formula with the ±π correction.
About the Inverse Trigonometric Functions Calculator
Each inverse trigonometric function returns one angle, its principal value: sin⁻¹ in [−π/2, π/2], cos⁻¹ in [0, π], tan⁻¹ in (−π/2, π/2), cot⁻¹ in (0, π), sec⁻¹ in [0, π] without π/2, and cosec⁻¹ in [−π/2, π/2] without 0.
So sin⁻¹(sin θ) equals θ only when θ is in [−π/2, π/2]; sin⁻¹(sin 5π/6) is π/6. tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)] when xy < 1, with π added (x > 0) or subtracted (x < 0) when xy > 1.
The formula
tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)], xy < 1
Principal values
| Function | Domain | Principal range |
|---|---|---|
| sin⁻¹ | −1 ≤ x ≤ 1 | [−π/2, π/2] |
| cos⁻¹ | −1 ≤ x ≤ 1 | [0, π] |
| tan⁻¹ | every real x | (−π/2, π/2) |
| cot⁻¹ | every real x | (0, π) |
| sec⁻¹ | |x| ≥ 1 | [0, π] except π/2 |
| cosec⁻¹ | |x| ≥ 1 | [−π/2, π/2] except 0 |
Worked example
Example: sin⁻¹(1/2) = π/6
- Inputs: I want: A principal value, like sin⁻¹(−√3/2); Function: sin⁻¹; x: 1/2.
- Formula: tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)], xy < 1
- sin⁻¹: the angle in [−π/2, π/2] whose sin is 1/2: sin⁻¹(1/2) = π/6
- Result: sin⁻¹(1/2) = π/6
- Principal range: [−π/2, π/2]
- Domain: −1 ≤ x ≤ 1
- In degrees: 30°
Questions people ask
Why is cos⁻¹(−1/2) = 2π/3 and not −π/3?
cos⁻¹ returns an angle in [0, π], and 2π/3 is the one there with cosine −1/2.
Is sin⁻¹x the same as 1/sin x?
No. sin⁻¹x is the inverse function (arcsin); 1/sin x is cosec x.