Arctan Calculator

Consider the right-angled triangle below.

In this triangle, the right angle is at B. Tan of the angle y is given by

$$tan\, y\, =\, {opposite \over adjacent} \,=\,{AB\over BC} = x$$

To find the angle y when the ratio AB/BC is known, we use the following formula for arctan.

$$y\, =\, tan^{-1}{opposite \over adjacent} \,= tan^{-1}\,{AB \over BC}\, = tan^{-1} x$$

The expression y = tan^-1⁡x, means that tan y = x , when -π/2 < y < π/2.

Arctan(x)=y is defined as the set of all angles whose tan is x. It should be noted here that the -1 is not an exponent and hence, tan^-1⁡x ≠ 1 / tan x .

Given below are some properties of the arctan function.

Angle – The angle is the parameter which is calculated when tan inverse (or arctan) of the angle is known.

Domain – The arctan of the parameter whose tan is x. This value can be any real number.

Range – The angles for which we can find tan^(-1)⁡x. This value is -π/2 < y < π/2 radians or -90 < y < +90 degrees.

Given below is a graph of the arctan function.

In the above graph, the y-axis denotes the angle in radians, and the x-axis denotes arctan of that angle. The red curve denotes the arctan function. For any value of x, the arctan function returns a value between -π/2 and π/2 radians (or -90⁰ to 90⁰).

For example, when x=1, y=arctan(x) returns y= π/4 radians. This means that tan (π/4) = 1.

If the arctan graph is compared to the tan graph, one would see that the arctan graph can be obtained from the tan graph by simply interchanging the horizontal and vertical axes.

Given below are some commonly used values of arctan(x).

The value of arctan(x) can be positive or negative, depending on the value of x. For example, arctan(-1) is negative, whereas arctan(1) is positive.

This is because tan^-1⁡(-x) = - tan^-1⁡x .

This has been illustrated below.

In this figure, the x-axis denotes the arctan values, whereas the y-axis denotes the angle θ. Arctan(x) will be positive when 0≤ θ < π/2 (in Quadrant 1), and negative when -π/2 < θ <0 (in Quadrant 4). Note that the angles - π/2 and π/2 are not included in this range.

We can also note that

tan(tan^-1⁡(x))=x when -∞ ≤ x ≤ +∞.

For example, tan(tan^-1⁡(1)) = tan 45 = 1.

Similarly, tan^-1⁡⁡(tan x) = x when -π/2 < x < +π/2.

For example, tan^-1⁡⁡(tan π/4) = tan^-1⁡⁡⁡(1) = π/4.

If we need to find the arctan of the tan of an angle which does not lie between -π/2 and π/2, the correct angle can be found by either adding or subtracting π radians until we get an angle in the range of -π/2 to π/2, which is the range of the arctan function. This is because π is the period of the tan function.

From a right-angled triangle, the arctan formula can be computed as

$$y = tan^{-1}{AB \over BC} = tan^{-1} \,x$$

where AB is the opposite side and AC is the adjacent side, and y is the angle which can be calculated.

The following series expansions can be used for the arctan function for different ranges of x.

Trigonometric functions as well as the inverse trigonometric functions are used widely in engineering, geometry and physics. They are also used to solve trigonometric equations describing real-life problems. The inverse tan function, in particular, is used quite extensively for finding the angles of elevation and depression, which is a crucial part of science and engineering.

In this section, we consider some example problems involving the use of the arctan formula.

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