Trigonometry & Mathematics

Tan Inverse Calculator (Arctan)

Q: What is the formula for calculating tan inverse (arctan)?

The tan inverse function, written as arctan(x) or tan⁻¹(x) , computes the angle θ whose tangent equals x: θ = arctan(x) where tan(θ) = Opposite ÷ Adjacent = x . The standard principal value range is -π/2 (or -90° ). Calculate angles in radians and degrees using our free Tan Inverse Calculator .

Q: What is tan inverse of 1 in radians and degrees?

The tan inverse of 1 is exactly π/4 radians (approximately 0.785398 rad ) or 45 degrees . Because tan(45°) = 1 / 1 = 1, arctan(1) = π/4 = 45°.

Q: What is tan inverse of -3 in radians and degrees?

The tan inverse of -3 is approximately -1.249046 radians or -71.565 degrees . Because arctan is an odd function ( arctan(-x) = -arctan(x) ), arctan(-3) = -arctan(3) = -1.2490 rad.

Q: What is tan inverse of 2 in radians and degrees?

The tan inverse of 2 is approximately 1.107149 radians or 63.435 degrees . In degrees and minutes, it equals 63° 26′ 6″.

Q: What is tan inverse of -1 in radians and degrees?

The tan inverse of -1 is exactly -π/4 radians (approximately -0.785398 rad ) or -45 degrees (or 315° in positive standard circle orientation).

Q: How do you convert tan inverse results from radians to degrees?

To convert any radian angle to degrees, multiply the radian value by (180 / π) : Degrees = Radians × (180 ÷ 3.14159265) . Conversely, to convert degrees to radians, multiply by (π ÷ 180) .

Q: What is the difference between atan and atan2?

The standard atan(x) function accepts a single ratio (y/x) and returns angles restricted to Quadrants I and IV (-90° to +90°). The atan2(y, x) function accepts separate Y and X Cartesian coordinates and returns the complete 360° angle (-180° to +180°), correctly distinguishing Quadrants II and III.

Free online tan inverse (arctan) calculator. Calculate inverse tangent in radians, degrees, and exact π fractions with step-by-step angle formulas.

Reviewed by Dr. Marcus Vance, Ph.D. (Ph.D. Applied Mathematics (MIT))

Verified for accuracy•Updated 2026

Interactive Calculator

1. Provide Details

Value (x)

Primary Display Unit

2. Output Results

Angle in Degrees (°)45.0000° °

Angle in Radians (rad)0.785398 rad rad

π Radian Fraction0.2500π rad

Trigonometric QuadrantQuadrant I (0° to 90°)

How to Calculate: Formula & Steps

Calculates the angle whose tangent is equal to x. For x = 1, arctan(1) = 45° = π/4 rad. For negative values, arctan(-x) = -arctan(x).

Formula Used:θ = arctan(x) = tan⁻¹(x); Degrees = Radians × (180 / π)

Step-by-Step Calculation Example

For x = 1: arctan(1) = 0.7854 rad = 45° = π/4. For x = -3: arctan(-3) = -1.2490 rad = -71.57°.

Common Mistakes to Avoid

Confusing tan⁻¹(x) with 1/tan(x) (cotangent). Inverse tangent finds an angle, whereas cotangent is the reciprocal ratio.
Forgetting to check whether your calculation expects radians or degrees.
Ignoring the standard principal range of arctan, which is strictly between -90° and +90° (-π/2 to π/2).

Practical Use Cases

Physics & engineering vector angle calculations
Roof pitch, slope incline, and ramp gradient measurements
Calculus integration and differential equations
Robotics coordinate transformations

Expert Tips

For x = 1, arctan is always exactly 45° (π/4 rad).
For very large positive values of x, arctan approaches +90° (π/2 rad).
Use atan2(y, x) if you are computing 2D vectors in Quadrants II or III.

Tan Inverse (Arctan) Formula, Unit Circle Values & Real-World Applications

In trigonometry, the inverse tangent — written arctan(x), atan(x), or tan⁻¹(x) — is the inverse operation of the tangent function. Given an input ratio x (opposite ÷ adjacent), it returns the angle θ whose tangent equals x: θ = arctan(x) where tan(θ) = x.

Because the tangent function repeats every 180° (π radians), the arctan function is restricted to a principal value range of (-π/2, π/2) or (-90°, 90°). Use our free Tan Inverse Calculator to compute angles in radians, degrees, and exact π fractions instantly.

Domain, Range & Mathematical Properties

Domain (Input x)

All Real Numbers (-∞, ∞)

Any real value is accepted as a tangent ratio.

Range (Output θ)

(-π/2, π/2) or (-90°, 90°)

Excludes exactly ±90° where tangent is undefined.

Odd Symmetry

arctan(-x) = -arctan(x)

Arctan is an odd function — negative inputs give negative angles.

Key Tan Inverse Values: Unit Circle Reference Table

Input Value (x)	Exact Radians	Decimal Radians	Degrees
-√3 ≈ -1.732	-π/3	-1.04720	-60°
-1	-π/4	-0.78540	-45°
-1/√3 ≈ -0.577	-π/6	-0.52360	-30°
0	0	0	0°
1/√3 ≈ 0.577	π/6	0.52360	30°
1	π/4	0.78540	45°
√3 ≈ 1.732	π/3	1.04720	60°
-3	≈ -1.24905	-1.24905	≈ -71.57°
2	≈ 1.10715	1.10715	≈ 63.43°

Converting Arctan Output: Radians ↔ Degrees

Scientific calculators, Python, JavaScript, and Excel all compute arctan in radians by default. Use these two formulas to convert between angular units:

Radians → Degrees

Degrees = Radians × (180 / π)

Example: 0.7854 × 57.2958 = 45°

Degrees → Radians

Radians = Degrees × (π / 180)

Example: 45° × 0.0174533 = 0.7854 rad

Math.atan(x) vs. Math.atan2(y, x) in Programming

In computer science and robotics, computing angles from coordinates using Math.atan(y / x) introduces quadrant ambiguity — division discards the sign information of both x and y, so it cannot distinguish between angles that are 180° apart.

The Math.atan2(y, x) function inspects the individual signs of y and x to return the correct angle across all four quadrants, from -180° to +180°:

atan(y/x)	atan2(y, x) Result	Quadrant
atan(1/1) = 45°	atan2(1, 1) = 45°	Quadrant I (+x, +y)
atan(-1/1) = -45°	atan2(-1, 1) = -45°	Quadrant IV (+x, -y)
atan(1/-1) = -45° (WRONG)	atan2(1, -1) = 135°	Quadrant II (-x, +y)
atan(-1/-1) = 45° (WRONG)	atan2(-1, -1) = -135°	Quadrant III (-x, -y)

Real-World Engineering, Physics & Construction Applications

Roof Pitch & Slope Angles: A 4/12 roof pitch has a rise of 4 inches per 12 inches of run. The angle is arctan(4/12) = arctan(0.333) ≈ 18.43°. Use this with our Roof Shingle Calculator to estimate materials.
ADA Wheelchair Ramp Compliance: The Americans with Disabilities Act requires a maximum 1:12 ramp slope, equivalent to arctan(1/12) ≈ 4.76°. At 1:20, the angle drops to ≈ 2.86°.
Electrical AC Power Factor (Phase Angle): An AC circuit's phase angle is θ = arctan(X / R), where X is reactance and R is resistance. This angle directly determines the power factor cos(θ).
Land Surveying & Grade Slopes: Elevation grade percentages convert to angles using arctan. A 6% grade means a 6-foot vertical rise per 100 horizontal feet: arctan(0.06) ≈ 3.43°.
Game Development & Camera Rotation: Game engines use atan2 to rotate sprites and calculate aim angles by comparing the player's position to the cursor position.

For a deeper dive into the arctan formula, odd/even symmetry, and the Taylor series expansion, read our full Tan Inverse Formula Guide.

Disclaimer

This calculation tool is provided for educational and informational estimation purposes only. Results are based on mathematical formulas and user-supplied parameters. They do not constitute formal underwriting, financial, tax, engineering, or legal determinations.

Frequently Asked Questions About Tan Inverse Calculator (Arctan)

What is the formula for calculating tan inverse (arctan)?

The tan inverse function, written as arctan(x) or tan⁻¹(x), computes the angle θ whose tangent equals x: θ = arctan(x) where tan(θ) = Opposite ÷ Adjacent = x. The standard principal value range is -π/2 < θ < π/2 (or -90° < θ < 90°). Calculate angles in radians and degrees using our free Tan Inverse Calculator.

What is tan inverse of 1 in radians and degrees?

The tan inverse of 1 is exactly π/4 radians (approximately 0.785398 rad) or 45 degrees. Because tan(45°) = 1 / 1 = 1, arctan(1) = π/4 = 45°.

What is tan inverse of -3 in radians and degrees?

The tan inverse of -3 is approximately -1.249046 radians or -71.565 degrees. Because arctan is an odd function (arctan(-x) = -arctan(x)), arctan(-3) = -arctan(3) = -1.2490 rad.

What is tan inverse of 2 in radians and degrees?

The tan inverse of 2 is approximately 1.107149 radians or 63.435 degrees. In degrees and minutes, it equals 63° 26′ 6″.

What is tan inverse of -1 in radians and degrees?

The tan inverse of -1 is exactly -π/4 radians (approximately -0.785398 rad) or -45 degrees (or 315° in positive standard circle orientation).

How do you convert tan inverse results from radians to degrees?

To convert any radian angle to degrees, multiply the radian value by (180 / π): Degrees = Radians × (180 ÷ 3.14159265). Conversely, to convert degrees to radians, multiply by (π ÷ 180).

What is the difference between atan and atan2?

The standard atan(x) function accepts a single ratio (y/x) and returns angles restricted to Quadrants I and IV (-90° to +90°). The atan2(y, x) function accepts separate Y and X Cartesian coordinates and returns the complete 360° angle (-180° to +180°), correctly distinguishing Quadrants II and III.