Arctan Calculator

Tangent value (x)

Angle

Find the angle from x and y coordinates (atan2)

Enter a point's x and y to get its four-quadrant angle from the positive x-axis.

Angle

deg (°)

Angle in radians (multiples of pi)
Sine of the angle (x over sqrt of 1 plus x squared)
Cosine of the angle (1 over sqrt of 1 plus x squared)

Complementary angle (arccot of x): deg (°)

The tangent of 45 deg (°) is 1, which confirms the inverse: arctan undoes tangent.

Arctan always returns the principal angle between -90 and 90 degrees (-pi/2 to pi/2 radians), even though every tangent value matches infinitely many angles.

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Show the arctan graph

Plot the inverse tangent S-curve and its horizontal asymptotes at plus and minus 90 degrees.

Show the angle in every unit

List your angle in degrees, radians, multiples of pi, gradians, and turns.

Your angle in every unit
Unit	Your angle
Degrees	45 deg
Radians	0.785398 rad
Radians (multiples of pi)	0.25 pi
Gradians	50 grad
Turns	0.125 turn

A tangent inverse calculator finds the angle whose tangent is equal to a given number. The tangent function takes an angle and gives a ratio. Conversely, the inverse tangent function does the opposite and converts any real number into its corresponding angle. You can either input a tangent value to retrieve its associated angle or enter an angle to find its tangent value. The unit of the angle can be changed at any time between degrees, radians, gradians, and revolutions. When coordinate mode is enabled, atan2 also determines the full angle in the four quadrants where a point is located.

What is the inverse tangent function?

The inverse tangent function, denoted as arctan(x), tan⁻¹(x) or atan(x), is the inverse function of the tangent function.

An inverse function reverses the operation done by a particular function. So if the tangent of an angle x is y, then the inverse tangent of y is x.

\tan(\theta) = x \quad\Longleftrightarrow\quad \arctan(x) = \theta

The arctangent function differs from similar functions like arcsine and arccosine in two ways. First of all, it accepts any real number as input. Since the range of tangent is negative infinity to positive infinity, the arctangent is not limited to a particular range such as -1 to 1, and will accept any value directly. Second, since the tangent repeats every 180 degrees, there are infinitely many angles that have the same tangent value. To give a single answer, the arctangent function chooses the principal value, which is an angle between -90 and 90 degrees (in radians: from -pi/2 to pi/2).

Note the notation "tan⁻¹(x)". This is an inverse function and not a reciprocal. The reciprocal of the tangent value, divided by 1, is the cotangent, another function.

How to use this calculator.

When you enter the value of tangent in the first input field, the corresponding angle is immediately shown. The result is clearly presented and also displayed in the Angle field where you can switch between degrees, radians, percent degrees and turns. The panel also shows the angle as a multiple of Pi, the sine and cosine of this angle, and the complementary angle.

This tool allows calculations in both directions. If you leave the Tangent field blank and enter an angle in the Angle field, you will get the value of the tangent for that angle. By activating coordinate mode and entering x- and y-coordinates, you can find the angle in all four quadrants. Opening the graph lets you see the entire arctangent curve while opening the table shows all units simultaneously for the angle.

Formula for arctangent:

This relationship can be expressed in a single line of code. The angle is the arctangent of x, and the value of the tangent of that angle equals x again.

\theta = \arctan(x) \qquad \tan(\theta) = x

Here is an example of a calculation: The value of the tangent of 45 degrees is 1, so the arctangent of 1 equals 45 degrees, which corresponds to pi/4 radians.

\tan(45^\circ) = 1 \quad\Longrightarrow\quad \arctan(1) = 45^\circ = \frac{\pi}{4}

Calculating an angle in a right-angled triangle.

If two sides of a right triangle are known, the arctangent is a standard method for calculating the angle. The tangent of an angle equals the ratio of the opposite side to the adjacent side, and this angle is the arctangent of that ratio.

\beta = \arctan\!\left(\frac{\text{opposite}}{\text{adjacent}}\right)

Suppose the opposite side is 10 and the adjacent side is 18. The ratio is found by dividing 10 by 18, which gives approximately 0.555. So the angle is the arctangent of 0.555, which is about 29 degrees.

\tan(\beta) = \frac{10}{18} \approx 0.555 \quad\Longrightarrow\quad \beta = \arctan(0.555) \approx 29.03^\circ

If only the opposite side and hypotenuse are known, then one can first use the Pythagorean theorem to calculate the adjacent side, then take the ratio, and finally determine the arctangent.

Slope, roof slope, road gradient

If the ratio between the vertical distance and horizontal distance is known, then the arctangent can be used to convert that slope into an angle. For a roof described as "6 in 12", the height increases by 6 units for every 12 units of horizontal extent, so the slope is the arctangent of the ratio of 6 to 12.

\text{angle} = \arctan\!\left(\frac{\text{rise}}{\text{run}}\right) = \arctan\!\left(\frac{6}{12}\right) \approx 26.57^\circ

The grade of a street can be calculated the same way and is usually expressed as a percentage. The grade is found by dividing the vertical distance by the horizontal distance and then multiplying by 100.

A grade of 100 percent corresponds to a slope of 1, which is an angle of 45 degrees, steeper than almost any street you will drive on. Surveyors, construction workers, and civil engineers rely on the arctangent to convert measured grades into angles that are indicated by a level.

atan2 and the four quadrants

The normal arctangent function can only take into account a ratio, which is why it cannot distinguish between the points (1, 1) and (-1, -1). Both have a tangent value of 1 but are separated by 180 degrees. The atan2(y, x) function with two arguments solves this problem by taking in the components y and x separately, considering their signs, and determining which quadrant the angle is in based on that.

\text{atan2}(y, x) = \text{the angle of the point } (x, y) \text{ from the positive } x\text{-axis}

The range of angles returned by the arctangent function is -90 to 90 degrees, while atan2 covers the full circle from -180 to 180 degrees. This function should be used when converting cartesian coordinates into polar coordinates, determining the orientation of a vector, or orienting objects in graphics or games towards other objects. Try it out with the coordinate mode above. Enter x and y values and you can see the angles for all four quadrants.

Commonly used arctangent values

There are some values that come up again and again. This table shows these values in degrees and radians, and if you enter a tangent value the calculator will also show one of these values. As this function is an odd function, for negative inputs only the sign of the angle changes.

Tangent value (x)	Angle in degrees	Angle in radians
-1.732	-60°	$-\frac{\pi}{3}$
-1	-45°	$-\frac{\pi}{4}$
-0.577	-30°	$-\frac{\pi}{6}$
0	0°	$0$
0.577	30°	$\frac{\pi}{6}$
1	45°	$\frac{\pi}{4}$
1.732	60°	$\frac{\pi}{3}$

As x approaches positive infinity, the angle approaches 90 degrees but never reaches it. As x approaches negative infinity, the angle approaches -90 degrees. Even if you enter a very large value like 1,000,000, the returned angle is already approximately 89.99994 degrees.

Inverse tangent graph

Plotting the inverse tangent for every value of tangent produces a gradually increasing S-shaped curve. It passes through the origin, rises steeply near zero, then becomes increasingly flat and approaches two horizontal asymptotes at 90 and -90 degrees (pi/2 and -pi/2 radians). These asymptotes correspond to the vertical asymptotes of the tangent function, which is a characteristic feature of inverse functions. The slope of any point on the curve is the derivative.

\frac{d}{dx}\arctan(x) = \frac{1}{1 + x^2}

Useful identities for the inverse tangent

There are some identities that make the inverse tangent more useful and connect it with other trigonometric functions.

\tan(\arctan(x)) = x

\sin(\arctan(x)) = \frac{x}{\sqrt{1 + x^2}} \qquad \cos(\arctan(x)) = \frac{1}{\sqrt{1 + x^2}}

\arctan(x) + \operatorname{arccot}(x) = \frac{\pi}{2}

The identity of complementary angles shows that the sum of an angle with a tangent value x and an angle with a cotangent value x is always 90 degrees. In addition, there is an addition formula which combines two inverse tangents into one value, as well as a formula which relates it to the inverse sine.

\arctan(\alpha) + \arctan(\beta) = \arctan\!\left(\frac{\alpha + \beta}{1 - \alpha\beta}\right)

\arctan(x) = \arcsin\!\left(\frac{x}{\sqrt{1 + x^2}}\right)

Applications of inverse tangent

The arctangent function is required in many fields since it can convert a ratio into an angle. In vector mathematics, it is used to find the orientation from x and y components. The atan2 function with two arguments computes this over all four quadrants and is therefore used in navigation, robotics, and computer graphics. Construction workers and surveyors use it to determine the slope angle from the ratio of elevation change to horizontal distance. Electricians calculate phase angles of AC circuits from the ratio of reactance to resistance, and in signal processing and control theory two components are similarly used to extract a phase.

Practical tips and hints for the arctangent function:

As the function for connection accepts any real numbers, it is not necessary to limit the range of values. Even if very large ratios are created in the calculation, the angle approaches only very strongly either the value 90 or -90 degrees.

The results are always quotient values and the range is between -90 and 90 degrees. To find the angle of another half wave that corresponds to the same tangent value, add or subtract 180 degrees.

If the x and y components are known and the actual orientation is desired, then use the function atan2(y,x). The regular arctangent function y/x will lose information about the quadrant.

Degrees are used for woodworking, surveying and everyday use while radians are used in analysis and programming. The units can be changed using the Angle switcher to convert between these two.

There are three values worth remembering: The arctangent of 0 is 0 degrees, the arctangent of 1 is 45 degrees and the arctangent of the square root of 3 is 60 degrees.

The tangent of x to the power of negative one is an inverse function and not the result of dividing 1 by the tangent. The reciprocal of the tangent is the cotangent.

Frequently asked questions

What is the domain of arctangent?

It is all real numbers. The arccosine and arcsine functions only accept inputs from -1 to 1. This is because the sine and cosine values never exceed this range. In contrast, the tangent function has a range of negative infinity to positive infinity, so its inverse accepts any input value. If very large values are entered, an angle will be returned that is very close to 90 or -90 degrees.

What is the difference between atan2 and regular arctangent?

atan2 takes the y and x components as two separate arguments and uses the signs to determine the correct quadrant. The normal arctangent only looks at the ratio of y divided by x, so it cannot distinguish between positive and negative combinations, and will always return an angle between -90 and 90 degrees. atan2 covers the full range from -180 to 180 degrees, which is essential when converting cartesian coordinates into polar coordinates.

Why is the output limited to -90 to 90 degrees?

Because the tangent repeats every 180 degrees, one ratio corresponds to infinitely many angles. To make the inverse function a suitable single-valued function, its output is restricted to one branch, that is, an angle between -90 and 90 degrees. This branch ensures that the curve remains continuous and the derivative is simple, namely 1 divided by 1 plus x squared.

Is tangent of x to the negative one equal to the result when you divide 1 by the tangent?

No, this notation is misleading. The tangent of x to the minus one power is the inverse function, also called arctangent, and converts a ratio into an angle. The result of dividing 1 by the tangent is the cotangent, another function. They appear similar only because they use imprecise notation.

How big is the arctangent of 1? Can you calculate the arctangent without a calculator?

The arctangent of 1 is 45 degrees or pi/4 radians because the tangent of 45 degrees is equal to 1. When it comes to commonly used values, you can write tan(y) = x to find the corresponding angle. The arctangent of 0 is 0 degrees and the arctangent of square root of 3 is 60 degrees. For more complex inputs, it's best to consult a table of trigonometric values or use a calculator.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Inverse trigonometric functions
Definitions, principal values, domains, identities, and the atan2 function for arctan and its siblings.
Math is Fun: Inverse Sine, Cosine, Tangent
Plain-language explanation of how the inverse trig functions undo sine, cosine, and tangent.