Last updated: July 31, 2026
Inverse Trigonometric Functions Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
For a real x, compute arcsin, arccos, arctan, arccot, arcsec, and arccsc in degrees whenever each function is defined.
Where:
- x=Input value(unitless)
- θ=Inverse trigonometric angle(degrees)
- 1/x=Reciprocal used by secant and cosecant inverses(unitless)
Worked Examples
Input inside the unit interval
x = 0.5 makes arcsin and arccos valid but arcsec and arccsc undefined.
- 1Check that 0.5 is inside [-1,1].
- 2Compute arcsin, arccos, arctan, and arccot.
- 3Because |0.5| < 1, arcsec and arccsc are marked undefined.
Input outside the unit interval
x = 2 makes reciprocal-based functions valid while arcsin and arccos become undefined.
- 1Notice that x = 2 lies outside [-1,1].
- 2Set arcsin and arccos outputs to 0 with definition flags No.
- 3Use 1/x = 0.5 to compute arcsec and arccsc.
Boundary input x = 1
At x = 1, all six inverse functions are defined in the real setting used here.
- 1Use x = 1 and note that it lies in both domain tests.
- 2Compute arcsin(1), arccos(1), arctan(1), and arccot(1).
- 3Because 1/x = 1, arcsec and arccsc are also defined.
Non-finite input warning
The combined calculator accepts only finite real numbers.
- 1Validation runs before any inverse function is computed.
- 2NaN or Infinity stops the calculation.
- 3All numeric outputs stay 0 and the calculator returns an error message.
Introduction
The inverse trigonometric functions calculator compares six inverse trig functions for the same real input. It shows the degree values when each function is defined and reports clear Yes or No flags for the functions with restricted domains. This makes it a practical study tool for understanding how inverse trig domains differ.
Why combine all six functions
Seeing all inverse trigonometric values for one input helps you compare domains, ranges, and relationships at a glance.
arcsin and arccos share the unit-interval domain.
arctan and arccot are always defined for real x.
arcsec and arccsc depend on the reciprocal condition.
The same input can be valid for some functions and invalid for others.
Definition flags remove ambiguity.
Degree outputs make comparisons easy.
Domain rules for each function
Each inverse function has its own real-valued domain condition.
arcsin requires -1 ≤ x ≤ 1.
arccos requires -1 ≤ x ≤ 1.
arctan accepts any real x.
arccot accepts any real x in this implementation.
arcsec requires |x| ≥ 1.
arccsc requires |x| ≥ 1.
Undefined functions return 0 in the numeric field and a separate Yes or No definition flag.
How the values are computed
The calculator uses JavaScript inverse trig functions and then converts the radian outputs into degrees.
arcsin uses Math.asin(x).
arccos uses Math.acos(x).
arctan uses Math.atan(x).
arccot uses Math.atan2(1, x).
arcsec uses Math.acos(1/x).
arccsc uses Math.asin(1/x).
Why definition flags matter
Some inputs produce valid outputs for only part of the inverse trig family, so the flags clarify which numbers are meaningful.
No flag means the numeric zero is only a placeholder.
Yes flag means the reported degree value is valid.
Flags make domain comparisons immediate.
They help prevent misreading undefined cases.
They are especially useful near x = ±1.
They keep the combined output easy to scan.
Understanding arccot
This calculator defines arccot(x) with atan2(1, x), giving a principal angle in degrees.
arccot is always defined for real x here.
x = 0 gives 90°.
Positive x gives an acute principal angle.
Negative x gives an obtuse principal angle.
It complements arctan for many positive inputs.
The atan2 form handles signs cleanly.
How to use the results
Use the combined display to identify which inverse function fits your problem and whether the input satisfies its real domain.
Compare values for the same input.
Check the Yes or No flags first.
Use arcsin or arccos for unit-interval data.
Use arctan or arccot for unrestricted ratios.
Use arcsec or arccsc when reciprocals are natural.
Verify special-angle cases quickly.
Common mistakes
The most common confusion comes from assuming every inverse trig function shares the same domain.
Treating undefined outputs as real values.
Ignoring the companion definition flags.
Confusing arcsec with arccos.
Confusing arccsc with arcsin.
Forgetting that arctan is always defined.
Mixing reciprocal-based and direct-value domains.
FAQs
Why do some outputs show 0 with a No flag?
Zero is used as a placeholder when that inverse function is undefined for the input. The Yes or No flag tells you whether the value is meaningful.
Which inverse trig functions are always defined here?
arctan(x) and arccot(x) are defined for all real finite x in this calculator.
When are arcsin and arccos defined?
They are defined for real inputs only when x lies between -1 and 1 inclusive.
When are arcsec and arccsc defined?
They are defined when the absolute value of x is at least 1.
How is arccot computed?
It uses atan2(1, x) and converts the result from radians to degrees.
Why are the outputs shown in degrees?
Degrees make it easier to compare the six inverse trig results side by side.
Can I use x = 1?
Yes. At x = 1, all six inverse trigonometric functions used here are defined.
What happens for NaN or Infinity?
The calculator rejects non-finite input and returns an error with zeroed numeric fields.