An inverse trig function runs a trig function backwards: you give it a ratio, and it returns an angle. Because sine, cosine and tangent repeat forever, infinitely many angles share each ratio, so every inverse function picks one standard answer, called the principal value. This calculator finds that value for all six inverses, shows it in degrees, in radians and, when the angle is special, as an exact multiple of π, and then lists every other angle that gives the same ratio.
How to use the inverse trig functions calculator
- Choose the function: arcsin, arccos, arctan, arccsc, arcsec or arccot.
- Enter x, either a number (
-0.5,2) or an expression such assqrt(3)/2or1/sqrt(2). - Choose whether the headline answer is in degrees or radians. Both are always listed.
- Read the principal value, the principal range it comes from, a check that plugs the answer back in, and the general solution. If x lies outside the function’s domain, the calculator explains why instead of returning a value.
Domains and principal ranges
The ranges below are the standard conventions used in most US textbooks:
| Function | Domain of x | Principal range |
|---|---|---|
| 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 real numbers | between −90° and 90°, endpoints excluded |
| arccsc x | x ≤ −1 or x ≥ 1 | −90° to 90°, excluding 0° |
| arcsec x | x ≤ −1 or x ≥ 1 | 0° to 180°, excluding 90° |
| arccot x | all real numbers | between 0° and 180°, endpoints excluded |
The three reciprocal inverses reduce to the main ones:
Worked example: arccos(−0.5)
1. Check the domain. −0.5 lies between −1 and 1, so arccos is defined.
2. Find the reference value. cos 60° = 1/2, so the angle we want has a reference angle of 60°.
3. Use the principal range. arccos must return an angle from 0° to 180°. Cosine is negative in Quadrant II, so the answer is 180° − 60° = 120°, which is 2π/3 ≈ 2.0943951024 radians.
4. List all solutions. Every θ = 120° + 360°k or θ = −120° + 360°k has cos θ = −0.5.
The same angle answers arcsec(−2), because arcsec(−2) = arccos(−1/2) = 120°.
Why inverse functions need a principal range
A function can only be reversed if each output comes from exactly one input. Sine fails that test on its full domain: sin 30° and sin 150° both equal 1/2. Mathematicians fix this by restricting sine to −90° through 90°, where it rises steadily from −1 to 1 and takes each value once. arcsin is the inverse of that restricted piece. Cosine is restricted to 0° through 180° and tangent to the open interval between −90° and 90°, for the same reason. Graphs of these restricted pieces and their mirror images appear in the inverse trig graphs tool.
The arccot, arcsec and arccsc conventions
The three main inverses are standardized, but the reciprocal ones are not. For arccot, most precalculus and calculus texts — and this calculator — use the range (0, π), which keeps the graph continuous. Some programming libraries and older texts use arctan(1/x), which gives (−π/2, π/2] and a different answer for negative x. A few calculus books also pick a different branch for arcsec and arccsc when x is negative, to simplify derivative formulas. If your class follows another convention, compare the general solution: the principal value may differ, but the set of all solutions is the same.
Practical uses
Inverse trig functions turn measurements into angles. If a ramp rises 1 foot over a 12-foot horizontal run, its angle is arctan(1/12) ≈ 4.76°. If a 20-foot ladder reaches 18 feet up a wall, it leans at arcsin(18/20) ≈ 64.16° from the ground. For full triangle solutions from sides and angles, try the right triangle calculator or the law of sines calculator.
Frequently asked questions
Is sin⁻¹(x) the same as 1/sin(x)?
No. sin⁻¹(x) means arcsin(x), the angle whose sine is x. The reciprocal 1/sin(x) is csc(x). For example, sin⁻¹(0.5) = 30°, while 1/sin(0.5 rad) ≈ 2.0858.
Why is arcsin(2) undefined?
Sine never goes above 1 or below −1, so no angle has a sine of 2. The domain of arcsin and arccos is −1 ≤ x ≤ 1. For arcsec and arccsc it is the opposite: x must be at least 1 in size.
What range does arccot use here?
This calculator returns arccot(x) in (0°, 180°), computed as 90° − arctan(x). That keeps arccot continuous, so arccot(−1) = 135°. Some software computes arctan(1/x) instead, which gives −45° for the same input.
How do I find every angle with a given sine, not just one?
Use the principal value α. For sine, θ = α + 360°k or 180° − α + 360°k; for cosine, θ = ±α + 360°k; for tangent, θ = α + 180°k, where k is any integer. The calculator lists these families for you.