Understanding Arcus Tangent
Arcus tangent — usually written arctan(x), tan⁻¹(x), or atan(x) — is the inverse of the tangent function. Given a ratio x, it answers the question "what angle has this tangent?" Because a single ratio could come from an angle in more than one quadrant, the plain single-argument arctan(x) is defined to always return a value between -90° and 90° (-π/2 to π/2 radians), no matter what x is.
The two-argument arctangent: atan2(y, x)
This calculator uses the more general and more practical form, atan2(y, x), which takes the vertical value y and horizontal value x separately instead of collapsing them into a single ratio first. Because it sees the sign of both numbers, atan2 can place the angle in the correct one of all four quadrants and returns a value across the full circle, from -180° to 180° (-π to π radians):
- x > 0, y > 0 — Quadrant I, angle between 0° and 90°
- x < 0, y > 0 — Quadrant II, angle between 90° and 180°
- x < 0, y < 0 — Quadrant III, angle between -180° and -90°
- x > 0, y < 0 — Quadrant IV, angle between -90° and 0°
When x is positive, atan2(y, x) gives exactly the same result as arctan(y/x). The difference only shows up when x is negative or zero, which is precisely where a plain arctan(y/x) calculation would give the wrong quadrant or fail outright from division by zero.
Degrees vs. radians
Radians are the natural unit in calculus and most programming math libraries; degrees are more familiar for everyday angles. Conversion: degrees = radians × 180/π, and radians = degrees × π/180. A full circle is 360° = 2π radians.
Common applications
- Finding the bearing or heading from one point to another in navigation, robotics, and game development
- Converting rectangular (x, y) coordinates to polar coordinates (radius and angle)
- Finding a missing angle in a right triangle from its opposite and adjacent sides (surveying, construction)
- Computing the slope angle of a line, ramp, or roof pitch from its rise and run
Frequently Asked Questions
What is the difference between arctan(x) and atan2(y, x)?
Why is x = 0 and y = 0 not allowed?
How do I convert the result between degrees and radians?
What is the hypotenuse value used for?
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 |