# atan2

## Syntax

``atan2(Y,X)``

## Description

example

````atan2(Y,X)` computes the four-quadrant inverse tangent (arctangent) of `Y` and `X`. If `Y` and `X` are vectors or matrices, `atan2` computes arctangents element by element.```

## Examples

### Four-Quadrant Inverse Tangent for Numeric and Symbolic Arguments

Compute the arctangents of these parameters. Because these numbers are not symbolic objects, you get floating-point results.

`[atan2(1, 1), atan2(pi, 4), atan2(Inf, Inf)]`
```ans = 0.7854 0.6658 0.7854```

Compute the arctangents of these parameters which are converted to symbolic objects:

`[atan2(sym(1), 1), atan2(sym(pi), sym(4)), atan2(Inf, sym(Inf))]`
```ans = [ pi/4, atan(pi/4), pi/4]```

### Limit of Four-Quadrant Inverse Tangent

Compute the limits of this symbolic expression:

```syms x limit(atan2(x^2/(1 + x), x), x, -Inf) limit(atan2(x^2/(1 + x), x), x, Inf)```
```ans = -(3*pi)/4 ans = pi/4```

### Four-Quadrant Inverse Tangent of Array Input

Compute the arctangents of the elements of matrices `Y` and `X`:

```Y = sym([3 sqrt(3); 1 1]); X = sym([sqrt(3) 3; 1 0]); atan2(Y, X)```
```ans = [ pi/3, pi/6] [ pi/4, pi/2]```

## Input Arguments

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Input, specified as a number, vector, matrix, array, or a symbolic number, array, function, or expression. If `Y` is a number, it must be real. If `Y` is a vector or matrix, it must either be a scalar or have the same dimensions as `X`. All numerical elements of `Y` must be real.

Input, specified as a number, vector, matrix, array, or a symbolic number, array, function, or expression. The function also accepts a vector or matrix of symbolic numbers, variables, expressions, functions. If `X` is a number, it must be real. If `X` is a vector or matrix, it must either be a scalar or have the same dimensions as `Y`. All numerical elements of `X` must be real.

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If X ≠ 0 and Y ≠ 0, then

`$\text{atan2}\left(Y,X\right)=\text{atan}\left(\frac{Y}{X}\right)+\frac{\pi }{2}\text{sign}\left(Y\right)\left(1-\text{sign}\left(X\right)\right)$`

Results returned by `atan2` belong to the closed interval `[-pi,pi]`. Results returned by `atan` belong to the closed interval `[-pi/2,pi/2]`.

## Tips

• Calling `atan2` for numbers (or vectors or matrices of numbers) that are not symbolic objects invokes the MATLAB® `atan2` function.

• If one of the arguments `X` and `Y` is a vector or a matrix, and another one is a scalar, then `atan2` expands the scalar into a vector or a matrix of the same length with all elements equal to that scalar.

• Symbolic arguments `X` and `Y` are assumed to be real.

• If `X = 0` and `Y > 0`, then `atan2(Y,X)` returns `pi/2`.

If `X = 0` and `Y < 0`, then `atan2(Y,X)` returns `-pi/2`.

If `X = Y = 0`, then `atan2(Y,X)` returns `0`.

## Alternatives

For complex `Z = X + Y*i`, the call `atan2(Y,X)` is equivalent to `angle(Z)`.