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atan

Symbolic inverse tangent

Description

example

P = atan(Z) returns the inverse tangent (arctangent) of the elements of Z. All angles are in radians.

  • For real values of Z, atan(Z) returns values in the interval [-pi/2,pi/2].

  • For complex values of Z, atan(Z) returns complex values with the real parts in the interval [-pi/2,pi/2].

P = atan(Y,X) returns the four-quadrant inverse tangent of the elements of Y and X. This syntax with two input arguments is the same as atan2(Y,X).

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

Examples

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Depending on its arguments, atan returns floating-point or exact symbolic results.

Compute the inverse tangent function for these numbers. Because these numbers are not symbolic objects, atan returns floating-point results.

P = atan([-1, -1/3, -1/sqrt(3), 1/2, 1, sqrt(3)])
P = 1×6

   -0.7854   -0.3218   -0.5236    0.4636    0.7854    1.0472

Compute the inverse tangent function for the numbers converted to symbolic objects. For many symbolic (exact) numbers, atan returns unresolved symbolic calls.

symP = atan(sym([-1, -1/3, -1/sqrt(3), 1/2, 1, sqrt(3)]))
symP = 

(-π4-atan(13)-π6atan(12)π4π3)

Use vpa to approximate symbolic results with floating-point numbers.

vpaP = vpa(symP)
vpaP = (-0.78539816339744830961566084581988-0.32175055439664219340140461435866-0.523598775598298873077107230546580.463647609000806116214256231461210.785398163397448309615660845819881.0471975511965977461542144610932)

Plot the inverse tangent function on the interval from -10 to 10.

syms x
fplot(atan(x),[-10 10])
grid on

Many functions, such as diff, int, taylor, and rewrite, can handle expressions containing atan.

Find the first and second derivatives of the inverse tangent function.

syms z
D1 = diff(atan(z),z)
D1 = 

1z2+1

D2 = diff(atan(z),z,z)
D2 = 

-2zz2+12

Find the indefinite integral of the inverse tangent function.

I = int(atan(z),z)
I = 

zatan(z)-log(z2+1)2

Find the Taylor series expansion of atan(z).

T = taylor(atan(z),z)
T = 

z55-z33+z

Rewrite the inverse tangent function in terms of the natural logarithm.

R = rewrite(atan(z),'log')
R = 

log(1-zi)i2-log(1+zi)i2

Input Arguments

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Tangent of angle, specified as a symbolic number, variable, expression, or function, or as a vector, matrix, or array of symbolic numbers, variables, expressions, or functions.

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-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

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Inverse Tangent

The inverse tangent is defined as

atan(Z)=i2log(1iZ1+iZ).

Four-Quadrant Inverse Tangent

If X ≠ 0 and Y ≠ 0, then

atan(Y,X)=atan(YX)+π2sign(Y)(1sign(X)).

Results returned by atan(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].

Version History

Introduced before R2006a

See Also

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