atan2

Symbolic four-quadrant inverse tangent

Syntax

P = atan2(Y,X)

Description

P = atan2(Y,X) computes the four-quadrant inverse tangent (arctangent) of Y and X.

Symbolic arguments X and Y are assumed to be real, and atan2(Y,X) returns values in the interval [-pi,pi].

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.

P = [atan2(1,1), atan2(pi,4), atan2(Inf,Inf)]

P = 1×3

    0.7854    0.6658    0.7854

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

P = [atan2(sym(1),1), atan2(sym(pi),sym(4)), atan2(Inf,sym(Inf))]

P = 
$$\left(\begin{array}{ccc}\frac{\pi }{4}& \mathrm{atan}\left(\frac{\pi }{4}\right)& \frac{\pi }{4}\end{array}\right)$$

Limit of Four-Quadrant Inverse Tangent

Compute the limits of this symbolic expression.

syms x
P_minusinf = limit(atan2(x^2/(1 + x),x),x,-Inf)

P_minusinf = 
$$-\frac{3\hspace{0.17em}\pi }{4}$$

P_plusinf = limit(atan2(x^2/(1 + x),x),x,Inf)

P_plusinf = 
$$\frac{\pi }{4}$$

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]);
P = atan2(Y,X)

P = 
$$\left(\begin{array}{cc}\frac{\pi }{3}& \frac{\pi }{6}\\ \frac{\pi }{4}& \frac{\pi }{2}\end{array}\right)$$

Input Arguments

Y — y-coordinates

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix | symbolic array

y-coordinates, specified as a symbolic number, variable, expression, or function, or as a vector, matrix, or array of symbolic numbers, variables, expressions, or functions. All numerical elements of Y must be real.

Inputs Y and X must either be the same size or have sizes that are compatible (for example, Y is an M-by-N matrix and X is a scalar or 1-by-N row vector). For more information, see Compatible Array Sizes for Basic Operations.

X — x-coordinates

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix | symbolic array

x-coordinates, specified as a symbolic number, variable, expression, or function, or as a vector, matrix, or array of symbolic numbers, variables, expressions, or functions. All numerical elements of X must be real.

Inputs Y and X must either be the same size or have sizes that are compatible (for example, Y is an M-by-N matrix and X is a scalar or 1-by-N row vector). For more information, see Compatible Array Sizes for Basic Operations.

More About

Four-Quadrant Inverse Tangent

If X ≠ 0 and Y ≠ 0, then

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

Results returned by atan2(Y,X) belong to the closed interval [-pi,pi]. Meanwhile, results returned by atan(Y/X) 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.

• 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).

Version History

Introduced in R2013a