The ArcTangent (Inverse Tangent) Function
ArcTangent returns the angle whose tangent equals the given value. arctan(x) = θ where tan(θ) = x. Domain: all reals, Range: (-π/2, π/2). Perfect for slope angles.
Why This Mathematical Concept Matters
Why: ArcTangent is used for slope angles, conversion from rectangular to polar coordinates, and atan2(y,x) for full quadrant resolution.
How: arctan(x) returns the unique angle in (-π/2, π/2) such that tan(θ) = x. Unlike arcsin/arccos, the domain is all reals — tangent never equals ±1 at boundaries.
- ●arctan(1) = 45° — the angle of a line with slope 1. arctan(x) gives the angle of any line with slope x.
- ●Use atan2(y, x) when you need the angle in all four quadrants; arctan(y/x) loses quadrant info.
- ●d/dx[arctan(x)] = 1/(1+x²) — the derivative is always positive and bounded.
Examples — Click to Load
Inverse Trig Value Breakdown
All Related Inverse Values
arcsin vs arccos (Related)
Calculation Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
arctan(x) = slope angle — a line with slope x makes angle arctan(x) with the x-axis.
— Khan Academy
arctan accepts all real inputs — unlike arcsin/arccos, the domain is unbounded.
— Paul's Notes
Key Takeaways
- • arctan(x) returns the angle whose tangent is x. Domain: all reals, Range: (-π/2, π/2)
- • arctan(0)=0°, arctan(1)=45°, arctan(√3)=60°, arctan(1/√3)=30°. arctan(-x) = -arctan(x) (odd function)
- • arctan(x) + arctan(1/x) = π/2 for x > 0. tan(arctan(x)) = x for all real x
- • atan2(y, x) in programming returns the full 360° angle using both coordinates — use when quadrant matters
- • arctan gives the slope angle: a line with slope m has angle arctan(m) with the x-axis
Did You Know?
How ArcTangent Works
arctan(x) finds the unique angle θ in (-π/2, π/2) such that tan(θ) = x. Unlike arcsin and arccos, arctan accepts any real input because tangent can output any real value.
atan2 in Programming
arctan(y/x) loses quadrant information when x < 0. atan2(y, x) uses both coordinates to return the angle in [-π, π]. Essential for converting Cartesian to polar coordinates correctly.
Slope Angle
A line y = mx + b has slope m. The angle this line makes with the positive x-axis is arctan(m). For m = 1, the angle is 45° (diagonal).
Asymptotic Behavior
As x → +∞, arctan(x) → 90°. As x → -∞, arctan(x) → -90°. The range is open (-π/2, π/2) — arctan never exactly reaches ±90°.
Expert Tips
Memorize Key Values
arctan(0)=0°, arctan(1)=45°, arctan(√3)=60°, arctan(1/√3)=30°. Use the Unit Circle Calculator.
Use atan2 for Full Circle
When converting (x,y) to angle, use atan2(y,x) not arctan(y/x). atan2 handles all quadrants. See this calculator for single-value arctan.
No Domain Restrictions
arctan works for any real input. Try arctan(1000) ≈ 89.94°. Great for numerical stability compared to arcsin/arccos.
Verify with tan(arctan(x))
Always check: tan(arctan(x)) = x. Use the Tangent Calculator to confirm.
Inverse Trig Calculator Comparison
| Feature | ArcTangent | ArcSine | ArcCosine |
|---|---|---|---|
| Domain | All reals | [-1, 1] | [-1, 1] |
| Range | (-π/2, π/2) | [-π/2, π/2] | [0, π] |
| atan2 alternative | Yes | No | No |
| Slope angle | Yes | No | No |
| Undefined for | Never | |x| > 1 | |x| > 1 |
| Odd/Even | Odd | Odd | Neither |
| At x=0 | 0° | 0° | 90° |
| Common use | Slope, atan2 | Opp/hyp | Adj/hyp |
Frequently Asked Questions
Why does arctan accept any number?
Tangent can output any real value (from -∞ to +∞), so arctan can accept any real input. Its range is restricted to (-π/2, π/2) for a unique inverse. arcsin and arccos are limited to [-1, 1] because sin and cos only output that range.
What is atan2?
atan2(y, x) returns the angle in [-π, π] using both coordinates. It preserves quadrant: atan2(1,1)=45°, atan2(1,-1)=135°, atan2(-1,-1)=-135°. arctan(y/x) would give 45° for both (1,1) and (-1,-1) — wrong for the second.
What is arctan(∞)?
arctan(x) → 90° (π/2) as x → +∞. The limit is approached but never reached since the range is (-π/2, π/2). In practice, arctan(1e10) ≈ 90°.
How is arctan used for slope?
A line with slope m makes angle θ = arctan(m) with the x-axis. m = 1 gives 45°, m = √3 gives 60°. The rise-over-run ratio is tan(θ).
What is arctan(x) + arctan(1/x)?
For x > 0, arctan(x) + arctan(1/x) = π/2. For x < 0, the sum is -π/2. This identity is useful for simplifying expressions.
Why is the derivative of arctan nice?
d/dx[arctan(x)] = 1/(1+x²) has no square roots or singularities. The integral ∫ dx/(1+x²) = arctan(x) + C is fundamental in calculus.
Can arctan output 90° or -90°?
No. The range is the open interval (-π/2, π/2). arctan(x) approaches ±90° as x → ±∞ but never equals it. This is because tan(±90°) is undefined.
When to use arctan vs atan2?
Use arctan when you have a single ratio (e.g., slope m). Use atan2(y, x) when you have Cartesian coordinates and need the full 360° angle with correct quadrant.
ArcTangent by the Numbers
Official & Educational Sources
Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. For full 360° angles from (x,y) coordinates, use atan2(y,x) in your programming environment.