Understanding Inverse Trigonometric Functions
Inverse trigonometric functions ("arc" functions) undo what sine, cosine, and tangent do: instead of taking an angle and returning a ratio, they take a ratio and return the angle that produces it. Because sin, cos, and tan are periodic — many angles share the same ratio — each inverse function is defined to return a single principal value from a restricted range, so the calculator always gives one well-defined answer rather than infinitely many.
Domains, ranges, and formulas
- arcsin(x) = sin⁻¹(x) — domain [-1, 1], principal range [-90°, 90°]
- arccos(x) = cos⁻¹(x) — domain [-1, 1], principal range [0°, 180°]
- arctan(x) = tan⁻¹(x) — domain: all real numbers, principal range (-90°, 90°)
- arccsc(x) = arcsin(1/x) — domain |x| ≥ 1, principal range [-90°, 90°] excluding 0°
- arcsec(x) = arccos(1/x) — domain |x| ≥ 1, principal range [0°, 180°] excluding 90°
- arccot(x) = 90° − arctan(x) — domain: all real numbers, principal range (0°, 180°)
This calculator evaluates the chosen function with JavaScript's built-in Math.asin, Math.acos, and Math.atan (which return radians), converts the result to degrees by multiplying by 180/π, and enforces each function's domain before computing — so an out-of-range input (like arcsin(2)) is rejected instead of silently returning a wrong number.
Degrees vs. radians
Both units describe the same angle: radians = degrees × π/180, and a full circle is 2π radians = 360°. This calculator reports both so you can plug the result directly into a geometry problem (degrees) or a calculus/physics formula (radians, since derivatives and Taylor series for trig functions assume radian input).
Frequently Asked Questions
What are the domain and range of arcsin, arccos, and arctan?
Why do arcsin and arccos require inputs between -1 and 1?
How are arccsc(x) and arcsec(x) calculated?
How is arccot(x) defined, and why does its range differ from arctan?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |