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 Diagram
This diagram updates when you calculate. The angle θ equals arctan(opposite ÷ adjacent).
Inverse Tangent Graph
The graph highlights the current input value. Hover for details.
What is Inverse Tangent?
Inverse tangent is a trigonometric function that returns the angle whose tangent equals a given number. The inverse tangent is written as tan−1(x), arctan(x), or atan(x). All 3 notations refer to the same inverse trigonometric function. 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, the tangent of angle θ equals the opposite side divided by the adjacent side. The inverse tangent reverses that operation: given the ratio opposite / adjacent, arctan returns the angle θ.
The domain of the inverse tangent function is all real numbers (−∞, +∞), and the range (principal value) is (−π/2, π/2) in radians or (−90°, 90°) in degrees. The inverse tangent is an odd function, meaning arctan(−x) = −arctan(x).
Interactive: Domain & Range of arctan(x)
Drag the slider to see how arctan maps any real number to an angle between −90° and 90°.
How to Calculate Inverse Tangent?
To calculate the inverse tangent, use the formula θ = arctan(x), where x is the tangent value and θ is the resulting angle.
3 steps summarize the tangent calculation process for the inverse of tangent:
- Identify the input value (x). The input is any real number—a decimal, a fraction, or the ratio opposite / adjacent from a right triangle.
- Apply the arctangent function. Use a scientific calculator, a programming language’s
atan()function, or Wolfram Alpha to compute tan−1(x). - Convert the result. Most math libraries return the answer in 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 see each calculation step in real time.
1Input: x = 1
2arctan(1) = 0.7854 rad
3Convert: 0.7854 × 180/π = 45°
How to Use This Inverse Tangent Calculator?
To use this inverse tangent calculator, enter a number in the input field, select your preferred angle unit, and press “Calculate arctan(x).”
This arc tan calculator supports 2 input methods:
Input Methods
1
Decimal Value
Enter any real number (e.g., 0.5, 1, −2.75). The calculator computes arctan of that decimal and returns the angle in degrees, radians, and gradians.
2
Opposite / Adjacent
Enter the lengths of the opposite and adjacent sides of a right triangle. The calculator divides opposite by adjacent and applies arctan to the result, matching how trigonometry works in the unit circle and Pythagorean theorem problems.
After pressing calculate, the result panel shows 4 output formats: degrees, radians, gradians, and π fraction. Each value has a copy button for easy use in MATLAB, Excel, or Google Sheets.
How to Use Inverse Tangent on a Calculator?
Press the “2nd” or “Shift” key on a scientific calculator (such as Casio or TI–84), then press the “tan” button to access tan−1.
4 steps cover the process on most scientific calculators:
- 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 “=”. The calculator returns the angle.
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.
3 manual methods exist for computing the inverse tangent without a calculator:
- Taylor series (Maclaurin series). The arctangent series approximation tool uses x − x3/3 + x5/5 − x7/7. More terms give higher precision. This method works best when |x| ≤ 1.
- Known angle values. Memorize common results: arctan(0) = 0°, arctan(1) = 45°, arctan(√3) = 60°, arctan(1/√3) = 30°. Use these as reference points.
- CORDIC algorithm. This hardware–level method computes arctangent through repeated angle rotations using only addition, subtraction, and bit–shifting—common in embedded systems and real–time processing.
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 S–shaped curve that passes through the origin and approaches −π/2 (−90°) as x → −∞ and π/2 (90°) as x → +∞.
3 key features define the arctangent graphical representation:
- Horizontal asymptotes at y = −π/2 and y = +π/2. The function never reaches ±90°.
- Odd symmetry. The graph reflects through the origin because arctan(−x) = −arctan(x).
- Monotonically increasing. The curve always rises from left to right. The arctangent derivative equals 1 / (1 + x2), which stays positive for all x.
Interactive: arctan(x) Graph
Hover over the graph to see the value at any point. The dashed lines mark the asymptotes at ±π/2.
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. Complete table of tangent and inverse tangent values for standard angles. Click any row to auto–calculate that value.
| x (Input) | arctan(x) Degrees | arctan(x) Radians | Fraction of π |
|---|---|---|---|
| −∞ | −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). All three notations describe the exact same mathematical function—the function that accepts a real number and returns the angle whose tangent equals that number.
Each notation appears in different contexts:
- tan−1(x) is the most common notation on scientific calculators, including Casio, TI–84, and Sharp models. The superscript −1 means “inverse function,” not “1 ÷ tan(x).” This distinction frequently confuses students—tan−1(x) returns an angle, whereas 1/tan(x) equals cot(x).
- arctan(x) is the standard notation in formal mathematics, especially in calculus and analysis. The prefix “arc” refers to the arc length on a unit circle that corresponds to the angle. Wolfram Alpha, many university textbooks, and Wikipedia use this form.
- atan(x) is the programming and software notation. JavaScript uses
Math.atan(x), Python usesmath.atan(x), MATLAB usesatan(x), and Excel/Google Sheets use=ATAN(value). The two–argument variantatan2(y, x)extends the range to (−π, π] for correct quadrant detection in navigation and robotics.
A common pitfall: the notation tan−1(x) does not mean the reciprocal of tangent. The reciprocal of tan(x) is cot(x) = cos(x) / sin(x), which is a completely different function. When you see tan−1 on a calculator or in a textbook, it always refers to the inverse tangent (arctangent) operation.
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, including Casio and TI–84 models. The superscript −1 indicates “inverse function,” not “1 / tan(x).” This notation dominates algebra and trigonometry textbooks.
Finding tan−1 of Negative Numbers
The tan−1 of negative numbers returns a negative angle. Because arctan is an odd function, arctan(−x) = −arctan(x). A negative input always maps to an angle between −90° and 0° (−π/2 and 0 radians).
2 examples of tan−1 of negative numbers:
- 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). This result means an angle of −45° has a tangent value of −1. On the unit circle, this angle corresponds to a point in the fourth quadrant where the opposite side and adjacent side have equal absolute lengths but opposite signs.
Interactive: Negative Value Explorer
Enter 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 calculations
Yes, tan−1 is the same as inverse tangent. Both terms describe the identical inverse trigonometric function that accepts a number and returns the angle whose tangent equals that number. The notation tan−1(x) is used on most calculators, while “inverse tangent” appears in written mathematics. Other equivalent names include arctan(x) and atan(x).
The tangent function takes an angle and returns a ratio, while the inverse tangent takes a ratio and returns an angle. Tangent (tan) accepts an angle θ as input and outputs opposite / adjacent. The inverse tangent (tan−1) accepts a numeric value x and outputs the angle θ where tan(θ) = x. The tangent domain is all real numbers except odd multiples of π/2. The inverse tangent domain is all real numbers (−∞, +∞).
The inverse tangent button sits above the “tan” key on a scientific calculator. Press “2nd” or “Shift” first, then press “tan” to activate the tan−1 function. On Casio calculators, the button is labeled SHIFT + tan. On TI–84, press 2ND + TAN. On a phone calculator, rotate to landscape mode to reveal the inverse tangent button.
Use the Taylor series expansion arctan(x) = x − x3/3 + x5/5 − x7/7 + … when |x| ≤ 1. For |x| > 1, use the identity arctan(x) = π/2 − arctan(1/x). Alternatively, refer to a tangent table of memorized standard angle values: arctan(0) = 0°, arctan(1) = 45°, arctan(√3) = 60°. The CORDIC algorithm offers another approach using only shifts and additions.
The inverse tangent of 1 is 45° (or π/4 radians, approximately 0.7854 radians). This result means tan(45°) = 1, which occurs in a right triangle where the opposite and adjacent sides have equal length. The value arctan(1) = 45° is one of the most frequently referenced values in trigonometry, algebra, and phase angle calculation in electronics.
Tan converts an angle into a ratio, while inverse tan converts a ratio into an angle. The tangent function maps degrees or radians to a real number. The inverse tangent (arctan) reverses this mapping. Tan has a periodic domain with excluded values, while inverse tan accepts every real number and always returns an angle in the range (−90°, 90°).
Yes, use the formula =ATAN(value) in Excel or Google Sheets to calculate inverse tangent. The ATAN function returns the angle in radians. To convert the result to degrees, use =DEGREES(ATAN(value)). For a 2–argument version that determines the correct quadrant, use =ATAN2(x_num, y_num).
Rotate your iPhone to landscape orientation to reveal the inverse tangent button. The iPhone calculator app in portrait mode shows basic operations only. In landscape mode, the scientific calculator layout appears with inverse trigonometric functions. Tap “2nd” to toggle between tan and tan−1, then enter the value.
Yes, both Excel and Google Sheets support =ATAN(value) for inverse tangent calculation. The function returns radians. Wrap it in =DEGREES() for degree output. Google Sheets also supports =ATAN2(x, y) for two–argument arctangent with correct quadrant detection. Both platforms accept cell references as input, making batch trigonometry calculations possible.
In a right triangle, the inverse tangent of the ratio opposite / adjacent equals the angle at the base of that triangle. The tangent function relates 2 sides of a right triangle to an angle θ: tan(θ) = opposite / adjacent. The inverse tangent reverses this: θ = arctan(opposite / adjacent). This relationship is central to solving right triangles using the Pythagorean theorem alongside inverse trigonometric functions.
There are 6 common real–life applications of arctangent:
- Navigation systems use arctan to convert cartesian coordinates to bearing angles.
- Phase angle calculation in electrical engineering uses arctan to find the phase shift between voltage and current.
- Computer graphics use atan2 to determine rotation angles for sprites and camera movement.
- Surveying and construction use arctan to compute slopes and elevation angles.
- Robotics use inverse tangent for joint angle calculations in kinematics.
- Signal processing uses arctan in frequency analysis and filter design.
Inverse tangent connects to sine and cosine through the identity tan(θ) = sin(θ) / cos(θ). When θ = arctan(x), sin(θ) = x / √(1 + x2) and cos(θ) = 1 / √(1 + x2). These relationships allow conversion between all 3 primary inverse trigonometric functions: arcsin, arccos, and arctan. Together, they form the complete set of inverse trigonometric functions used in calculus for integration and differentiation.
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.