An inverse tangent calculator (also known as an arcus tangent calculator) computes the arctangent (atan) of any real number and returns the angle whose tangent equals that value. Whether you are finding the angle of a slope by entering rise and run as opposite and adjacent, or simply converting a tangent ratio to degrees or radians, this angle finder gives you the result instantly. Enter a decimal or the ratio of opposite / adjacent sides from a right triangle, choose between degrees and radians, and get instant results. This arc tan calculator also draws an interactive triangle diagram, plots the arctan(x) graph, and displays a full tangent table—everything needed for trigonometry, calculus, and navigation calculations in one place.
Inverse Tangent Calculator
Result
Calculated arctan(1)
45°
Degrees 45°
Radians 0.7854 rad
Gradians 50 grad
π Fraction π/4
Interactive Right Triangle
After each calculation, the triangle updates so you can see that θ = arctan(opposite / adjacent).
Inverse Tangent Graph
The graph marks the current input so you can compare the number with its angle.
What is Inverse Tangent?
Inverse tangent, written as tan⁻¹(x), arctan(x), or atan(x), returns the angle whose tangent equals x. In European and German textbooks this same function appears as arcus tangens or arkus tangens. On the unit circle, arctan returns the angle that corresponds to any tangent value along the circle.
In a right triangle, tan(θ) = opposite / adjacent, so arctan(opposite / adjacent) gives the angle θ.
Its domain is all real numbers, and its principal range is (-π/2, π/2) or (-90°, 90°).
Interactive: Domain and Range of arctan(x)
Move the slider to see how any real input maps to an angle between -90° and 90°.
How to Calculate Inverse Tangent?
Use θ = arctan(x) when x is a decimal value or a ratio.
- Identify the value x.
- Use a scientific calculator or an atan() function to compute arctan(x).
- Convert the result if you need degrees instead of radians. Multiply the radian result by 180/π to get degrees, or multiply a degree value by π/180 to convert back to radians. Either way the inverse tan formula remains θ = arctan(x) throughout.
Interactive: Step-by-Step Calculation
Type a value to update each step instantly.
1Input: x = 1
2arctan(1) = 0.7854 rad
3Convert: 0.7854 × 180/π = 45°
How to Use This Inverse Tangent Calculator?
Choose a decimal or opposite/adjacent input, select degrees or radians, and press Calculate.
Input Methods
1
Decimal Value
Decimal mode accepts values such as 0.5, 1, or -2.75.
2
Opposite / Adjacent
Opposite / Adjacent mode divides the 2 sides first and then applies arctan.
The result panel shows multiple answer formats for quick use.
How to Use Inverse Tangent on a Calculator?
Press the “2nd” or “Shift” key on a scientific calculator (like Casio or TI-84), then press the “tan” button to access tan⁻¹.
- Turn on the calculator and set the angle mode to degrees (DEG) or radians (RAD).
- Press “2nd” or “Shift.” This key activates the secondary functions printed above each button. This shift tan key combination is universal across Casio, TI-84, and Sharp models. Always set your angle mode to DEG or RAD before pressing it, since the same input returns different numbers depending on the mode.
- Press the “tan” key. The display shows “tan⁻¹(” or “atan(”.
- Type the value and press “=”.
Visual Guide: Finding tan⁻¹ on a Calculator
The image below shows the exact buttons to press on a TI-84 scientific calculator.
Finding tan⁻¹(1) = 45° on a TI-84 calculator
How to Calculate Inverse Tangent Without a Calculator?
Use the arctangent Taylor series expansion to approximate arctan(x) by hand: arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + … This series converges when |x| ≤ 1.
- Taylor Series (Maclaurin series). The arctangent series approximation tool uses x − x³/3 + x⁵/5 − x⁷/7. More terms give higher precision.
- Known Angle Values. Memorize common results: arctan(0) = 0°, arctan(1) = 45°, arctan(√3) = 60°.
- CORDIC Algorithm. This hardware-level method calculates arctangent through repeated angle rotations.
Interactive: Taylor Series Approximation
Choose how many terms to include. Watch how the approximation converges toward the true value.
Inverse Tangent Graph
The arctan(x) graph is an increasing S-shaped curve that passes through the origin and approaches ±π/2.
- It has horizontal asymptotes at y = -π/2 and y = π/2.
- It is odd, so arctan(-x) = -arctan(x).
- It always increases from left to right.
Interactive: arctan(x) Graph
Hover over the graph to inspect exact values.
Tangent Table
The tan inverse values at 0°, 30°, 45°, 60°, and 90° appear in nearly every trigonometry course and are worth committing to memory. Click any table row to load that standard value into the calculator.
| x | arctan(x) Degrees | arctan(x) Radians | π Fraction |
|---|---|---|---|
| −∞ | −90° | −1.5708 rad | −π/2 |
| −√3 ≈ −1.7321 | −60° | −1.0472 rad | −π/3 |
| −1 | −45° | −0.7854 rad | −π/4 |
| −1/√3 ≈ −0.5774 | −30° | −0.5236 rad | −π/6 |
| 0 | 0° | 0 rad | 0 |
| 1/√3 ≈ 0.5774 | 30° | 0.5236 rad | π/6 |
| 1 | 45° | 0.7854 rad | π/4 |
| √3 ≈ 1.7321 | 60° | 1.0472 rad | π/3 |
| +∞ | 90° | 1.5708 rad | π/2 |
Notation for the Inverse of Tangent
The notation for the inverse of tangent takes 3 standard forms: tan⁻¹(x), arctan(x), and atan(x).
- tan⁻¹(x) Common on physical calculators.
- arctan(x) Standard in formal mathematics.
- atan(x) Programming and software notation.
Some students informally call it anti tangent since it reverses what tangent does, though this is not standard mathematical terminology.
Interactive: Notation Comparison
Click each notation to highlight where it appears in math, programming, and calculators.
tan⁻¹(x) is the most common notation on physical calculators. The superscript -1 indicates “inverse function,” not “1 / tan(x)”.
Finding tan⁻¹ of Negative Numbers
The tan⁻¹ of negative numbers returns a negative angle. Because arctangent is an odd function, arctan(-x) = -arctan(x).
- arctan(-1) = -45° or -π/4 radians
- arctan(-√3) = -60° or -π/3 radians
What is the tan⁻¹ of -1?
The tan⁻¹ of -1 is -45° (-π/4 radians or -0.7854 radians).
Interactive: Negative Value Explorer
Input any negative number to see how arctan maps negative inputs to negative angles.
arctan(-1) = -45° = -π/4 rad
arctan(1) = 45° ↔ arctan(-1) = -45°
FAQs
Answers to common questions about inverse tangent.
Yes, tan⁻¹ is the standard mathematical notation for inverse tangent, also called arctangent or arctan. The superscript ⁻¹ does not mean 1/tan (which is cotangent). Instead, tan⁻¹(x) asks: 'What angle has a tangent equal to x?' For example, tan⁻¹(1) = 45° because tan(45°) = 1. You may also see it written as arctan(x) or atan(x) in programming languages.
The tangent function takes an angle and returns a ratio of the opposite side to the adjacent side in a right triangle. The inverse tangent (arctan) does the reverse: it takes a ratio and returns the corresponding angle. For instance, tan(45°) = 1, so arctan(1) = 45°. Tangent has a domain of all real numbers except odd multiples of 90°, while inverse tangent accepts all real numbers and outputs angles between −90° and 90° (−π/2 to π/2 radians).
The inverse tangent button sits above the tan key on most scientific calculators. To access it, press the 2nd or Shift key first, then press the tan button. The display will show tan⁻¹ or arctan. On graphing calculators like the TI-84, press 2nd then TAN. On Casio calculators, press SHIFT then tan.
Use the Taylor series expansion: arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + ... for |x| ≤ 1. For common values, memorize key angles: arctan(0) = 0°, arctan(1/√3) = 30°, arctan(1) = 45°, arctan(√3) = 60°. You can also use the CORDIC algorithm. For values outside [−1, 1], use the identity arctan(x) = π/2 − arctan(1/x) for x > 0.
The inverse tangent of 1 is 45° (or π/4 radians). This is because tan(45°) = 1. In a right triangle where the opposite and adjacent sides are equal, the angle is 45°.
Tan converts an angle into a ratio (opposite/adjacent), while inverse tan (arctan) converts a ratio back into an angle. Tan is periodic and can produce any real number, while arctan always returns a unique angle in the range (−90°, 90°).
Yes, use the formula =ATAN(value) in Excel. This returns the result in radians. To convert to degrees, use =DEGREES(ATAN(value)). Excel also supports =ATAN2(x, y) for two-argument arctangent.
Rotate your iPhone to landscape orientation to reveal the scientific calculator. Tap the 2nd button to switch to inverse functions. The tan button will change to tan⁻¹. Enter your value and tap tan⁻¹.
In a right triangle, the inverse tangent of the ratio opposite/adjacent equals the angle at that vertex. If opposite = 3 and adjacent = 4, then arctan(3/4) ≈ 36.87°.
There are 6 common applications: (1) Navigation and GPS for bearing angles, (2) Engineering for slope angles, (3) Physics for force vectors, (4) Computer graphics for rotation, (5) Electrical engineering for phase angles, (6) Astronomy for elevation angles.
Through tan(θ) = sin(θ)/cos(θ). If θ = arctan(x), then sin(θ) = x/√(1+x²) and cos(θ) = 1/√(1+x²). The derivative of arctan(x) is 1/(1+x²).
Yes. A slope ratio is the same as opposite over adjacent in a right triangle. Enter your rise as the opposite side and your run as the adjacent side and the tool returns the angle of inclination in both degrees and radians.
This tool is built specifically for arctangent. If you need a full inverse trig calculator covering arcsin and arccos alongside arctan, you would need a broader inverse trigonometry tool.