Math
Inverse Trig Functions Calculator
Inverse trig functions answer the question "which angle has this sine (or cosine, tangent...)?" Enter the value and mathcheck returns the principal angle in degrees and radians, shows the exact π fraction when it exists, lists every angle in [0°, 360°) with that value, writes the general solution, and explains domain errors such as arcsin 2.
Decimals, fractions, and radicals all work: 0.5, 1/2, √3/2, -sqrt(2)/2. arcsin and arccos need -1 ≤ x ≤ 1; arccsc and arcsec need |x| ≥ 1.
Try an example
Result
Principal value
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- All solutions in [0°, 360°)
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- General solution
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- Radians
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- Range of the function
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Study path
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What this calculator solves
Each trig function repeats, so many angles share one value. The inverse function picks a single principal angle from an agreed range. This calculator gives that principal value and also the other angle in one full turn, because homework problems usually ask for every solution.
Domain and range of each inverse function
| Function | Domain (allowed x) | Range (angle returned) | Second solution in one turn |
|---|---|---|---|
| arcsin x | -1 ≤ x ≤ 1 | -90° to 90° | 180° − θ |
| arccos x | -1 ≤ x ≤ 1 | 0° to 180° | 360° − θ |
| arctan x | all real x | -90° to 90° (open) | θ + 180° |
| arccsc x | |x| ≥ 1 | -90° to 90°, θ ≠ 0° | 180° − θ |
| arcsec x | |x| ≥ 1 | 0° to 180°, θ ≠ 90° | 360° − θ |
| arccot x | all real x | 0° to 180° (open) | θ + 180° |
How to use it
- Choose the inverse function you need. Use the The number you get by flipping a fraction; 3/4 becomes 4/3, and 5 becomes 1/5. ones (arccsc, arcsec, arccot) when the problem gives csc, sec, or cot.
- Type the value. A root expression written with the √ symbol, such as a square root. like √3/2 are recognized so the answer can be exact.
- Read the principal value in degrees and An angle unit where a full turn is 2π instead of 360 degrees., then the full list of solutions from 0° to 360°.
- Use the general solution line when the problem asks for all real solutions.
How to read the answer
- A principal value like 30° = π/6 is exact; one like 36.87° is rounded to four decimals.
- arcsin and arctan of a negative number are negative angles; arccos of a negative number is an obtuse angle.
- The unit-circle figure shows the principal angle in blue and the other solution in teal.
Common mistakes and edge cases
- arcsin 2 and arccos 1.5 are undefined because sine and cosine never leave [-1, 1].
- arccsc 0.5 is undefined because cosecant never lies strictly between -1 and 1.
- sin⁻¹ x means arcsin x, not 1 / sin x. The reciprocal of sine is cosecant.
- Calculators only return the principal value; if your angle is in One of the four regions the x- and y-axes cut the coordinate plane into. II or III you must use 180° − θ or θ + 180° yourself.
- arccot conventions differ: this calculator returns angles between 0° and 180°, and the warning shows the atan(1/x) alternative when they disagree.
Worked examples
arcsin 1/2
Principal value
30° = π/6
arccos of a negative value
arccos(-√3/2) is obtuse
Principal value
150° = 5π/6
arctan of a decimal
arctan 0.75 (a 3-4-5 triangle)
Principal value
36.8699° = 0.6435 rad
arctan(-1)
principal value is negative
Principal value
-45° = -π/4
arccsc 2
Principal value
30° = π/6
arccot(-1)
uses the 0° to 180° convention
Principal value
135° = 3π/4
arcsin 0
zero solutions merge at 0° and 180°
Principal value
0° = 0
Out of domain
arcsin 2 is undefined
Principal value
undefined
arcsec 0.5
secant never lies between -1 and 1
Principal value
undefined
Frequently asked questions
Why is arcsin 2 undefined?+
Sine is the y-coordinate of a point on the unit circle, so it is always between -1 and 1. No angle has sine 2, and the calculator reports a domain error instead of a number.
What is the difference between sin⁻¹ x and 1/sin x?+
sin⁻¹ x (arcsin x) is the inverse function that returns an angle. 1/sin x is the reciprocal, which is cosecant. The -1 in sin⁻¹ is notation for the inverse, not an exponent.
Why does arcsin 1/2 give 30° when 150° also has sine 1/2?+
A function can return only one output, so arcsin is defined to return angles from -90° to 90°. The calculator also lists 150° under All solutions because most trig equations want every angle in one turn.
How do I get the answer in radians?+
The radian value is shown next to the degrees. Special angles appear as π fractions (π/6, 5π/6, 3π/4); other angles show a decimal such as 0.6435 rad.
Which range does arccot use?+
This calculator returns arccot values between 0° and 180°, matching most textbooks. Some software uses atan(1/x), which gives a negative angle for negative inputs; the result panel notes the alternative whenever they differ.
How do I solve sin θ = 0.5 for all θ?+
Take the principal value 30°, find the second solution 180° − 30° = 150°, then add multiples of 360° to both: θ = 30° + 360°k or θ = 150° + 360°k.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026