# arctan

# What is Arctan and How to Calculate It?

Arctan is one of the inverse trigonometric functions, also known as the inverse tangent function. It is used to find the angle that has a given tangent ratio. In other words, it is the function that “undoes” the tangent function.

In this article, we will explain what arctan means, how to calculate it using a calculator or a formula, and how to use it in real-life applications.

## Arctan Definition

The arctan of x is defined as the inverse tangent function of x when x is real (x ââ). When the tangent of y is equal to x:

`tan y = x`

Then the arctan of x is equal to the inverse tangent function of x, which is equal to y:

`arctan x = tan x = y`

For example:

`arctan 1 = tan 1 = Ï/4 rad = 45Â°`

The arctan function can also be written as atan or tan, but we will use the arc- prefix notation throughout this article.

## Arctan Calculator

The easiest way to calculate arctan is to use an online calculator, such as this one. You just need to enter the tangent value, select degrees (Â°) or radians (rad) and press the = button. The calculator will display the arctan value in both units.

For example, if you want to calculate arctan(0.5), you can enter 0.5 in the calculator and select degrees. The calculator will show that arctan(0.5) = 26.565Â°.

## Arctan Formula

If you don’t have access to a calculator, you can also use a formula to calculate arctan. However, this method is more complicated and requires some knowledge of trigonometry and calculus.

The formula for arctan is derived from the Taylor series expansion of the inverse tangent function:

`arctan x = x - x/3 + x/5 - x/7 + ...`

This formula works for any value of x, but it converges very slowly when |x| > 1. Therefore, it is more efficient to use some trigonometric identities to reduce the value of x before applying the formula.

For example, if you want to calculate arctan(2), you can use the following identity:

`arctan x = Ï/2 - arctan(1/x)`

This identity works for any nonzero value of x, but it is especially useful when |x| > 1. By applying this identity, we can reduce the value of x from 2 to 0.5:

`arctan 2 = Ï/2 - arctan(1/2)`

Now we can use the Taylor series formula to calculate arctan(0.5):

`arctan 0.5 â 0.5 - 0.5/3 + 0.5/5 - 0.5/7 + ...`

If we truncate the series after four terms, we get:

`arctan 0.5 â 0.5 - 0.041666667 + 0.008333333 - 0.002380952 â 0.463648`

This value is in radians. To convert it to degrees, we multiply by 180/

`arctan 0.5 â 0.463648 Ã 180/Ï â 26.565Â°`

Now we can use the identity to find arctan(2):

`arctan 2 = Ï/2 - arctan(0.5) â Ï/2 - 0.463648 â 1.107149`

This value is also in radians. To convert it to degrees, we multiply by 180/Ï again:

`arctan 2 â 1.107149 Ã 180/Ï â 63.435Â°`

This is the same result as the calculator gave us.

## Arctan Applications

The arctan function has many applications in various fields of science and engineering. Here are some examples:

- In geometry, arctan can be used to find the angle of a right triangle given the opposite and adjacent sides, or the slope of a line given two points.
- In physics, arctan can be used to find the angle of incidence or reflection of a light ray, or the angle of launch or impact of a projectile.
- In navigation, arctan can be used to find the bearing or direction of a destination given the coordinates of the current location and the destination.
- In astronomy, arctan can be used to find the angular size or distance of a celestial object given its actual size and distance.
- In electronics, arctan can be used to find the phase angle of an alternating current or voltage given its real and imaginary components.

## Conclusion

In this article, we have learned what arctan is, how to calculate it using a calculator or a formula, and how to use it in real-life applications. We hope you have found this article helpful and informative.