Formula and Method for the Tan Inverse Calculator
The inverse tangent, written tan⁻¹(x) or arctan(x), is the angle θ whose tangent equals x. If tan(θ) = x, then θ = tan⁻¹(x). Since tan(θ) = opposite / adjacent in a right triangle, tan⁻¹(x) answers the question "what angle produces this ratio of opposite to adjacent side?" This calculator takes any real number x — positive, negative, zero, or very large — and returns the principal angle θ in degrees, radians, and gradians, plus a check that recomputes tan(θ) so you can confirm it matches your input.
How the calculation works
Enter x, the ratio you want the angle for. The calculator computes θ = tan⁻¹(x) in radians using the arctangent function, then converts that radian value to degrees (θ° = θ_rad × 180/π) and gradians (θ_grad = θ_rad × 200/π). Because tangent repeats every 180°, tan⁻¹ always returns its principal value, strictly between -90° and 90° (-π/2 and π/2 radians) — it never returns 90° or -90° exactly, since tan(θ) is undefined at those points. The tool also shows tan(θ) recomputed from the result, which should match your original x (subject to rounding) as a sanity check.
arctan(x) vs. atan2(y, x)
tan⁻¹(x) only sees the ratio, so it cannot tell whether the original opposite and adjacent sides were both positive or both negative — 1/1 and (-1)/(-1) both give x = 1 and the same 45° answer, even though those correspond to different quadrants. If you need the full angle for a coordinate pair (x, y), including which quadrant it lies in across the whole -180° to 180° range, use the two-argument function atan2(y, x) instead, which examines the sign of each side separately.
Common applications
- Finding an unknown angle in a right triangle when the opposite and adjacent side lengths (or their ratio) are known.
- Computing the angle of a slope, ramp, or roof pitch from its rise-over-run ratio.
- Determining the direction of a 2D vector or line segment in physics, engineering, and computer graphics.
- Evaluating derivatives and integrals involving tan⁻¹(x) in calculus, where d/dx[tan⁻¹(x)] = 1/(1 + x²).
Frequently Asked Questions
What is the difference between arctan(x) and atan2(y, x)?
What is the range of the inverse tangent function?
How do I convert the arctan result between degrees and radians?
What is tan⁻¹(1) and why?
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 |