Inverse Trigonometric Calculator — All 6 Functions
Find the angle for all six inverse trigonometric functions: arcsin, arccos, arctan, arccsc, arcsec, and arccot. Enter a value and select a function to get the principal angle in both degrees and radians.
Function
Principal value — range: [−90°, 90°]
- 1
arcsin(x) in radians
arcsin(0.5) = 0.523599Principal-value angle in radians. - 2
Convert to degrees
0.523599 × 180 ÷ π = 30
How does this calculator work?
All six inverse trig functions in one place: arcsin and arccos need x ∈ [−1, 1]; arctan and arccot accept any real number; arccsc and arcsec need |x| ≥ 1. The reciprocal inverses are computed via arccsc(x) = arcsin(1/x), arcsec(x) = arccos(1/x), arccot(x) = π/2 − arctan(x). Results are in degrees and radians.
Formula
How this is calculated
The six inverse trigonometric functions are the inverses of sine, cosine, tangent, cosecant, secant, and cotangent. Because all six trig functions are periodic, their inverses are defined only on restricted principal ranges that make them single-valued. Arcsin and arccos require inputs in [−1, 1] (the range of sine and cosine); arctan and arccot accept any real number; arccsc and arcsec require |x| ≥ 1 (since sine and cosine are bounded by ±1, their reciprocals cosecant and secant have magnitude ≥ 1).
The three reciprocal inverses are computed from the primary ones: arccsc(x) = arcsin(1/x), arcsec(x) = arccos(1/x), and arccot(x) = π/2 − arctan(x). The last identity holds under the convention that arccot has principal range (0°, 180°), which is the most widely used convention in analysis and physics. Radian results are converted to degrees by multiplying by 180/π.
A key point about uniqueness: each inverse function returns exactly one angle from its principal range, even though infinitely many angles satisfy the same trig equation due to periodicity. For example, sin(30°) = sin(150°) = 0.5, but arcsin(0.5) = 30° only. To find other solutions, use the general solutions: θ = arcsin(x) + 360°·k or θ = 180° − arcsin(x) + 360°·k for integer k.
Frequently asked questions
Cosecant and secant are the reciprocals of sine and cosine, which are always in [−1, 1]. Their reciprocals therefore have magnitude ≥ 1, so values strictly between −1 and 1 are outside the range of csc and sec.
Under the principal range convention (0°, 180°), arccot(x) = 90° − arctan(x) for all real x. This is equivalent to atan(1/x) only when x > 0; for x < 0, atan(1/x) falls in (−90°, 0°) while arccot(x) falls in (90°, 180°).
Because trig functions repeat every 360°, each value has infinitely many angles satisfying the equation. The principal value is the single "canonical" angle chosen from a standard restricted range — the range that makes the inverse function continuous and unique.
Also known as
TG we-Calculate Editorial Team. (2026). Inverse Trigonometric Calculator — All 6 Functions [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/inverse-trigonometric-calculator
TG we-Calculate Editorial Team. "Inverse Trigonometric Calculator — All 6 Functions." TG we-Calculate. 2026. https://we-calculate.com/calculator/inverse-trigonometric-calculator.
TG we-Calculate Editorial Team, "Inverse Trigonometric Calculator — All 6 Functions," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/inverse-trigonometric-calculator
@misc{wecalculate_inverse_trigonometric_calculator, title = {Inverse Trigonometric Calculator — All 6 Functions}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/inverse-trigonometric-calculator}}, year = {2026}, note = {TG we-Calculate} }
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