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This tool covers all six inverse trigonometric functions in one place — not just the commonly-taught arcsin, arccos, and arctan, but also the less commonly available arccsc, arcsec, and arccot, defined via the reciprocal and complementary-angle relationships. Scientific calculators and spreadsheet software typically only include the first three as built-in buttons or functions, so anyone needing arccsc, arcsec, or arccot directly (rather than manually taking a reciprocal first) usually needs a dedicated tool like this one.
How it works
Choose which inverse function you need and enter a value. arcsin and arccos require a value between -1 and 1 inclusive, since sine and cosine themselves never exceed that range; arccsc and arcsec require |value| ≥ 1, since they're computed as arcsin(1/x) and arccos(1/x), and that reciprocal must land within [-1, 1]; arctan and arccot accept any real number at all, since tangent and cotangent are unbounded. For arccot specifically, this calculator uses the arccot(x) = 90° − arctan(x) convention, which produces a continuous output range of 0° to 180° that matches most standard textbook definitions, rather than an alternative convention that jumps discontinuously at x = 0.
- Enter inverse function.
- Enter value (x).
- Click Calculate to see your results.
Examples
arccot(1)
arccot(1) = 90° − arctan(1) = 90° − 45° = 45°.
arcsec(2)
arcsec(2) = arccos(1/2) = arccos(0.5) = 60°.
arccsc(2)
arccsc(2) = arcsin(1/2) = arcsin(0.5) = 30°.
Who should use it
- Finding an angle from a secant, cosecant, or cotangent ratio without manually taking a reciprocal first.
- Trigonometry and pre-calculus coursework covering all six inverse functions.
Industry applications
- Trigonometry and calculus education
- Engineering calculations involving less-common trig ratios
Advantages
- Covers all six inverse trig functions in one consolidated tool.
- Handles the domain restrictions for each function automatically.
Limitations
- arccot's 0°-180° convention used here is standard but not universal — some calculus texts define it differently, so cross-check the convention your course uses.
Common mistakes to avoid
- Entering a value strictly between -1 and 1 for arccsc or arcsec, which has no valid result since their defining reciprocal would fall outside arcsin/arccos's valid domain.
- Confusing arccot(x) with the reciprocal 1/arctan(x) — they are different operations that only coincidentally agree in special cases.
- Assuming all six inverse functions share the same output range — they don't, and mixing up ranges can produce a plausible-looking but wrong angle.
Best practices
- When in doubt about a less-common inverse function, rewrite it via its reciprocal relationship (e.g., arcsec(x) = arccos(1/x)) to double-check the result by hand.
- Keep track of which output range applies to the function you're using, especially when chaining results into further calculations.
Tips
- Need just arcsin, arccos, or arctan individually? Use the Arcsin, Arccos, or Arctan Calculator directly.