Use this arctan calculator khổng lồ quickly find the inverse tangent. Whether you"re looking for a simple answer khổng lồ the question "what is an arctan?" or are curious about the integral or derivative of arctan, you"ve come to the right place. Below, you"ll also find the arctan graph, as well as a neat table with commonly used values, such as arctan(1) & arctan(0). Alternatively, you can simply type the value of interest into this tool and you"ll find the answer in the blink of an eye.

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Interested in more advanced trigonometry? check out our law of sines and law of cosines calculators if you have an triangles khổng lồ solve.

Arctangent is the inverse of the tangent function. Simply speaking, we use arctan when we want khổng lồ find an angle for which we know the tangent value.

However, in the strictest sense, because the tangent is a periodic trigonometric function, it doesn"t have an inverse function. Still, we can define an inverse function if we restrict the domain to the interval where the function is monotonic. The commonly chosen interval, -π/2 Abbreviation Definition tên miền of arctan x Range of usual principal values arctan(x) tan-1x, atan x = tan(y) all real numbers R -π/2 -90°

Using the tan-1x convention may lead to lớn confusion about the difference between arctangent & cotangent. It turns out that arctan & cot are really separate things:

cot(x) = 1/tan(x), so cotangent is basically the reciprocal of a tangent, or, in other words, the multiplicative inversearctan(x) is the angle whose tangent is x

We hope that now you vày not doubt that arctan and cotan are different. Lớn avoid any further misunderstandings, you may want to lớn use the arctan(x) rather than tan-1x notation.

By restricting domain name of the principal tangent function, we obtain the inverse tangent that ranges from −π/2 to lớn π/2 radians exclusively. However, the domain of an arctangent function is all real numbers. The graph then looks as follows:

Graph Commonly used values
x arctan(x)
-∞ -π/2 -90°
-3 -1.2490 -71.565°
-2 -1.1071 -63.435°
-√3 -π/3 -60°
-1 -π/4 -45°
-√3/3 -π/6 -30°
0 0
√3/3 π/6 30°
1 π/4 45°
√3 π/3 60°
2 1.1071 63.435°
3 1.2490 71.565°
π/2 90°

How is this arctan graph created? By reflecting the tan(x) in the (-π/2 π/2) range through the line y = x. You can also look at it as swapping the horizontal & vertical axes:

## Arctan properties, relationships with trigonometric functions, integral & derivative of arctan

The relationships in trigonometry are crucial to lớn understanding this topic even more thoroughly. Inspecting the right-angled triangle with side lengths 1 và x is a good starting point if you want khổng lồ find the relationships between arctan and the basic trigonometric functions:

Tangent: tan(arctan(x)) = x

Other useful relationships with arctangent are:

arctan(x) = π/2 - arccot(x)arctan(-x) = -arctan(x)integral of arctan: ∫arctan(x) dx = x arctan(x) - (1/2) ln(1 + x²) + Carctan(x) + arctan(1/x) = π/2, for x > 0 & arctan(x) + arctan(1/x) = -π/2, for x

It"s easy to prove the first equation from the properties of the right triangle with side lengths 1 & x, as we perfectly know that the sum of angles in a triangle equals 180°. Subtracting the right angle, which is 90°, we"re left with two non-right angles, which must sum up to lớn 90°. Thus, we can write the angles as arctan(x) & arctan(1/x).

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This is one of our easiest calculators to lớn use, really! Just enter the number which you want khổng lồ find the arctan of. As the domain name of arctan is all real numbers, you don"t need to lớn worry too much. Let"s say we want khổng lồ find the arctan 1. Just type in the number and the inverse tangent calculator will display the result. As we expected, the arctan of 1 is equal to 45°. This arctan calculator works the other way round as well, that is as a standard tangent calculator - type the angle into the second box và tangent of that angle will appear.