# Plot $\arg(z)$ in an Argand diagram and display the angle

I'd like to ask you about the way to show the $$\arg(z)$$ annotation about the angle. My point is to show $$\frac{2\pi}{3}$$ and $$\frac{5\pi}{6}$$ on the image.

What I mean:

Is there any way to achieve similar effect in Mathematica?

My code:

    Show[%6, Axes -> True, Method -> {"ScRegionPlot[
Pi/4. <= Arg[x + I y] <= Pi, {x, -1, 1}, {y, -.1, 1},
GridLines -> {{-.15}, {0}},
PlotPoints -> 60, Show[%6,Axes\[Rule]True,Method\[Rule]{" \
ScRegionPlot[
Norm[x + I y + .15] < 0.6 &&
Pi/4. <= Arg[x + I y] <= Pi, {x, -1, 1}, {y, -.1, 1},
AspectRatio ->
Automatic] alingFunctions "\[Rule]None," \
TransparentPolygonMesh "\[Rule]True}]
AspectRatio -> Automatic]alingFunctions" -> None,
"TransparentPolygonMesh" -> True}] roots = z /. Solve[z^3 == 1, z];
pts = {Re[#], Im[#]} & /@ roots;
args = Arg /@ Cases[roots, _?(Im[N[#]] != 0 &)];

Module[{pr = 1.25, radius = {1/4, 3/8}},
Graphics[{
Thick,
Arrow[{{-pr, 0}, {pr, 0}}],
Arrow[{{0, -pr}, {0, pr}}],
Text[Style[Subscript[z, 0], 18, Bold],
pts[], {0, -2}],
Text[Style[Subscript[z, 1], 18, Bold],
pts[], {0, -2}],
Text[Style[Subscript[z, 2], 18, Bold],
pts[], {0, 2}],
{
Circle[{0, 0}, #[], {0, #[]}],
Text[Style[#[], 14, Bold], #[]*
Through[{Cos, Sin}[#[]/2]],
{-1.25, -Sign[#[]]}]} & /@
AbsolutePointSize,
Point[pts],
AbsoluteDashing[{10, 10}],
Line[{{0, 0}, #}] & /@ pts},
Axes -> True,
Ticks -> {Range[-1, 1, 1/2], Range[-1, 1, 1]},
TicksStyle -> Directive["Label", 14],
PlotRange -> {{-pr, pr}, {-pr, pr}},
AxesLabel -> (Style[#, 14, Bold] & /@ {Re, Im})]] A question on plotting complex numbers begs for an answer that directly uses complex numbers rather than requires pulling the complex numbers apart into real and imaginary parts. Such an answer is allowed by David Park's Presentations( add-on (https://home.comcast.net/~djmpark/DrawGraphicsPage.html):

   <<Presentations

roots = z /. Solve[z^3 == 1, z];
args = Arg /@ Select[roots, ! Element[#, Reals] &];

With[{rng = 1.25, radii = {1/4, 3/8}, arcOffset = 1.4},
Draw2D[{
Thick,
(*arrow axes *)
ComplexArrow[{-rng, rng}], ComplexArrow[{-rng I, rng I}],

(* labeled arcs *)
ComplexCircle[0, First@#, {0, Last@#}] & /@
ComplexText[Last@#,
ComplexPolar[ arcOffset First@#, Last@#/2], {-1, 0}] & /@

(* lines to roots *)
Dashing[0.035],
ComplexLine[{0, #}] & /@ Rest@roots,

(* labeled roots *)
PointSize[0.025], ComplexPoint /@ roots,
ComplexText[Sequence @@ #] & /@
Transpose[{Style[#, 18] & /@ {Subscript[z, 0], Subscript[z, 2],Subscript[z, 1]},
roots, {{0, -2}, {0, 2}, {0, -2}}}]
},
PlotRange -> rng,
Axes -> True, AxesLabel -> {Re, Im},
Ticks -> {Range[-1, 1, 1/2], Range[-1, 1, 1]},
TicksStyle -> Directive["Label", 14, FontWeight -> "Medium"],
BaseStyle -> Directive[14, Bold]
]
] Here's the same thing decorated a bit with use of color, etc.:

   With[{rng = 1.25, radii = {1/4, 3/8}, arcOffset = 1.4},
Draw2D[{
Thick,
(*arrow axes *)
{Gray, ComplexArrow[{-rng, rng}], ComplexArrow[{-rng I, rng I}]},

(* labeled arcs *)
{Legacy@SeaGreen, ComplexCircle[0, First@#, {0, Last@#}] & /@
ComplexText[Last@#,
ComplexPolar[ arcOffset First@#, Last@#/2], {-1, 0}] & /@

(* lines to roots *)
{Dashing[0.035], Legacy@BrownOchre,
ComplexLine[{0, #}] & /@ Rest@roots},

(* labeled roots *)
ComplexCirclePoint[#, 5, Black, Legacy@CadmiumOrange] & /@ roots,
ComplexText[Sequence @@ #] & /@
Transpose[{Style[#, 18] & /@ {Subscript[z, 0], Subscript[z, 2],Subscript[z, 1]},
roots, {{0, -2}, {0, 2}, {0, -2}}}]
},
PlotRange -> rng,
Axes -> True, AxesLabel -> {Re, Im},
Ticks -> {Range[-1, 1, 1/2], Range[-1, 1, 1]},
TicksStyle -> Directive["Label", 14, FontWeight -> "Medium"],
BaseStyle -> Directive[14, Bold],
Background -> Lighter@Legacy@Linen,
PlotLabel -> "The cube roots of unity"
]
]
` 