Inverse trigonometric functions: Difference between revisions

Revision as of 10:01, 4 September 2024

Revision as of 15:30, 4 September 2024

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{{anchor|Two-argument variant of arctangent}}

{{main|atan2}}

The two-argument [[atan2]] function computes the arctangent of ''y'' / ''x'' given ''y'' and ''x'', but with a range of (−{{pi}},&nbsp;{{pi}}]. In other words, atan2(''y'',&nbsp;''x'') is the angle between the positive ''x''-axis of a plane and the point (''x'',&nbsp;''y'') on it, with positive sign for counter-clockwise angles (upper half-plane, ''y''&nbsp;>&nbsp;0), and negative sign for clockwise angles (lower half-plane, ''y''&nbsp;<&nbsp;0). Equivalently, this is the the [[principal value]] of the [[arg (mathematics)|arg]]ument of the [[complex number]] ''x''&nbsp;+&nbsp;'''i'''''y''. atan2It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.

It is related to the arctangent of ''y'' / ''x'' but, when {{math|''x'' < 0}}, the angle <math>\arctan(y / x)</math> is [[antipodal point|diametrically opposite]] the desired angle, and ±{{pi}} (a half [[turn (angle)|turn]]) must be added to place the point in the correct [[quadrant (plane geometry)|quadrant]]; the result has a range of (−{{pi}},&nbsp;{{pi}}]. In addition, atan2 can be computed for {{math|1=''x'' = 0}} without dividing by zero.

More formally, that is how atan2 can be expressed inIn terms of the standard '''arctan''' function, that is with range of (−{{sfrac|{{pi}}|2}}, {{sfrac|{{pi}}|2}}), it can be expressed as follows:

:<math>\operatorname{atan2}(y, x) = \begin{cases}

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\text{undefined} & \quad y = 0,\; x = 0

\end{cases}</math>

It also equals the [[principal value]] of the [[arg (mathematics)|arg]]ument of the [[complex number]] ''x''&nbsp;+&nbsp;'''i'''''y''.

This limited version of the function above may also be defined using the [[tangent half-angle formula]]e as follows: