Angle

Command Summary

Computes the angle of a complex number or a set of complex numbers.

Command Syntax

angle(Expr1)expression
angle(List1)list
angle(Matrix1)matrix

The angle( command computes the angle of a complex number, which corresponds to the rotation of a vector whose length is equivalent to the abs( of the number. The angle is returned in degrees, gradians, or radians, depending on the mode the calculator is in.

Degree mode
:angle(1+i) = 45
:angle(1+i) = π/4

angle( can also return the angles for a list or matrix of complex numbers, returning the outputs in a list or matrix respectively. angle( will attempt to return a closed-form expression for the value rather than a decimal expression when utilizing the CAS, and can also return a symbolic evaluation of an angle.

:angle(z) = π(sign(z)-1)/2

# Formulas

The general formula for the angle of complex number x+iy is given by

(1)
\begin{align} \mathrm{angle}(x+iy) = \dfrac{\pi \times \mathrm{sign}(y)}{2} - \tan^{-1}\bigg(\dfrac{x}{y}\bigg) \end{align}

Often denoted as θ, the angle of a complex number is used in its polar representation

(2)
\begin{align} x+iy = re^{i\theta} \end{align}

where r is the absolute value of the number.