Choose arcsin, arccos, arctan, arccot, arcsec, or arccsc, enter a ratio, and get the angle back in both degrees and radians — with domain errors clearly flagged.
Choose a function and enter a ratio
This inverse trig calculator works backward compared to a typical trigonometry tool: instead of taking an angle and returning a ratio, it takes a ratio and returns the angle. Choose arcsin, arccos, arctan, arccot, arcsec, or arccsc from the dropdown, enter your known ratio, and this inverse sine cosine tangent calculator instantly returns the angle in both degrees and radians. Each of the six inverse functions has its own valid input domain — arcsin and arccos only accept ratios between -1 and 1, while arcsec and arccsc only accept ratios with absolute value 1 or greater — and this calculator checks that domain for you, alerting you clearly rather than returning a meaningless result when your ratio falls outside it.
Given a function choice and a ratio x, this calculator validates that x falls within that function's domain, computes the angle in radians using JavaScript's built-in Math.asin, Math.acos, or Math.atan (with arccot, arcsec, and arccsc derived from those three), and converts the radian result to degrees for display alongside the original radian value.
Precalculus and trigonometry students solving "find the angle" problems, physics students recovering a launch or incidence angle from velocity or refractive-index ratios, engineering and robotics students computing joint or bearing angles from position ratios, and anyone who needs an arcsin calculator in degrees (or any of the other five inverse functions) without deriving the domain rules from memory.
Forward trig functions (sin, cos, tan, cot, sec, csc) answer "given this angle, what's the ratio?" Inverse trig functions answer the opposite question — "given this ratio, what's the angle?" — which is exactly the question that comes up whenever you measure a slope, a ratio of distances, or a ratio of velocities and need to recover the underlying angle. Because each forward function repeats its values infinitely often, every inverse function is defined with a specific principal-value range so it returns exactly one angle per valid input.
Navigators and pilots use arctan to compute a bearing from north/east displacement components. Physicists use arcsin to find the critical angle for total internal reflection and the refraction angle in Snell's law. Architects and builders use arctan to find a roof pitch or staircase angle from a rise/run ratio. Robotics engineers use inverse trig functions throughout inverse-kinematics calculations to recover joint angles from target positions. Electrical engineers use arctan to find a circuit's phase angle from a reactance/resistance ratio.
How each inverse function is computed, and its valid input domain
Each inverse function has a defined principal-value range so it always returns exactly one angle, never multiple.
Arcsin/arccos need -1 to 1; arcsec/arccsc need |x| ≥ 1; arctan/arccot accept any real number.
Every result is shown in both units, since textbooks favor degrees while code and physics formulas favor radians.
From choosing a function to reading the angle result
Select Arcsin, Arccos, Arctan, Arccot, Arcsec, or Arccsc from the dropdown.
Type the numeric ratio (x-value) you want the angle for.
The calculator checks the domain for your chosen function and computes the angle in radians.
If your ratio falls outside the valid domain, an alert explains why and no result is shown.
Both units are displayed together in the results panel.
Finding the angle whose sine is 0.5, using Arcsin
A student measured a sine ratio of 0.5 in a physics experiment and needs to know the angle it corresponds to.
Explanation: 30° is one of the standard reference angles, and its sine (0.5) is a clean, easy-to-remember value — a useful check to confirm arcsin is working correctly before using it on a less familiar ratio.
Domain and principal-value range for all six inverse functions
| Function | Domain | Principal-Value Range |
|---|---|---|
| Arcsin(x) | -1 ≤ x ≤ 1 | -90° to 90° |
| Arccos(x) | -1 ≤ x ≤ 1 | 0° to 180° |
| Arctan(x) | All real numbers | -90° to 90° (exclusive) |
| Arccot(x) | All real numbers | 0° to 180° (exclusive) |
| Arcsec(x) | |x| ≥ 1 | 0° to 180°, excluding 90° |
| Arccsc(x) | |x| ≥ 1 | -90° to 90°, excluding 0° |
Reading the result: the calculator always returns the single principal-value angle for your chosen function — if you need a different coterminal angle with the same ratio, add or subtract the function's period manually.
Typical ranges: arcsin and arccsc results fall in [-90°, 90°]; arccos and arcsec results fall in [0°, 180°]; arctan and arccot results fall in the same ranges but never reach the exact endpoints.
Manual verification: take the returned angle and plug it back into the original forward function (sin, cos, tan, cot, sec, or csc) using a scientific calculator — it should reproduce the ratio you started with.
Where recovering an angle from a known ratio genuinely matters
Find a heading angle from north/east displacement components using arctan.
Recover a launch angle from vertical and horizontal velocity components.
Compute an incline angle from a measured rise-over-run ratio.
Solve for joint angles from target end-effector position ratios.
Find a refraction angle or the critical angle for total internal reflection using arcsin.
Back out an elevation or bearing angle from measured horizontal and vertical distances.
Recover a rotation angle from vector or coordinate components.
Find the phase angle in an AC circuit from a reactance-to-resistance ratio.
Determine a truss member or support angle from known geometry ratios.
Recover an observation angle from a ratio of distances or heights.
Solve "find the angle" problems across all six inverse functions.
Confirm a sin/cos/tan/cot/sec/csc calculation by working backward from the ratio.
Back-calculate an elevation angle from a known geometric ratio.
What this inverse trig calculator does well, and where it doesn't apply
Domain and principal-value range side by side
| Function | Domain | Range (degrees) | Range (radians) |
|---|---|---|---|
| Arcsin(x) | -1 ≤ x ≤ 1 | -90° to 90° | -π/2 to π/2 |
| Arccos(x) | -1 ≤ x ≤ 1 | 0° to 180° | 0 to π |
| Arctan(x) | All reals | -90° to 90° (excl.) | -π/2 to π/2 (excl.) |
| Arccot(x) | All reals | 0° to 180° (excl.) | 0 to π (excl.) |
| Arcsec(x) | |x| ≥ 1 | 0° to 180°, ≠90° | 0 to π, ≠π/2 |
| Arccsc(x) | |x| ≥ 1 | -90° to 90°, ≠0° | -π/2 to π/2, ≠0 |
Summary: This inverse trig calculator gives you an instant, free way to find the angle behind any arcsin, arccos, arctan, arccot, arcsec, or arccsc ratio, in both degrees and radians, with domain checks built in. Pair it with the forward-direction Cot, Sec, and Cosec calculators to cover angle-to-ratio conversions too.
Common questions about arcsin, arccos, arctan, and the other inverse trig functions
Trusted educational references to go deeper on inverse trig functions
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