直交座標表示と極座標表示

準備：パッケージの読み込み

<< Graphics`Arrow`

２つのベクトルの合成ベクトル（cos θ）

n = 64;
For[i = 0, 
      i < n, i++, 
θ = 
        2 Pi*i/n + 0.001; 
a = 
        E^(I θ)/2; 
b = 
        E^(-I θ)/2; 
c = 
        a + b; 
Show[
        Graphics[{
{RGBColor[0.5, 0.5, 0.5], 
              Circle[{0, 0}, 1]}, 
{RGBColor[0.5, 0.5, 
                0.5], Circle[{0, 0}, 1/2]}, 
{RGBColor[0.5, 
                0.5, 0.5], Dashing[{0.02, 0.02}], 
              Line[{{Re[a], Im[a]}, {Re[c], 
                    Im[c]}}]}, 
{RGBColor[0.5, 0.5, 0.5], 
              Dashing[{0.02, 0.02}], 
              Line[{{Re[b], Im[b]}, {Re[c], 
                    Im[c]}}]}, 
{RGBColor[1, 0, 0], 
              Arrow[{0, 0}, {Re[b], Im[b]}]}, 
{RGBColor[0, 
                0, 1], Arrow[{0, 0}, {Re[a], 
                  Im[a]}]}, 
{PointSize[0.04], 
              Point[{Re[c], Im[c]}]}
}, 
          PlotRange -> {{-1.3, 1.3}, {-1.3, 1.3}}, 
          AspectRatio -> Automatic, Ticks -> None, 
          ImageSize -> {200, Automatic}, 
          Axes -> Automatic]]
]

２つのベクトルの合成ベクトル（sin θ）

n = 64;
For[i = 0, 
      i < n, i++, 
θ = 
        2 Pi*i/n + 0.001; 
a = 
        E^(I θ)/
          2; 
b = -E^(-I θ)/
          2; 
c = a + b; 
Show[
        Graphics[{
{RGBColor[0.5, 0.5, 0.5], 
              Circle[{0, 0}, 1]}, 
{RGBColor[0.5, 0.5, 
                0.5], Circle[{0, 0}, 1/2]}, 
{RGBColor[0.5, 
                0.5, 0.5], Dashing[{0.02, 0.02}], 
              Line[{{Re[a], Im[a]}, {Re[c], 
                    Im[c]}}]}, 
{RGBColor[0.5, 0.5, 0.5], 
              Dashing[{0.02, 0.02}], 
              Line[{{Re[b], Im[b]}, {Re[c], 
                    Im[c]}}]}, 
{RGBColor[1, 0, 0], 
              Arrow[{0, 0}, {Re[b], Im[b]}]}, 
{RGBColor[0, 
                0, 1], Arrow[{0, 0}, {Re[a], 
                  Im[a]}]}, 
{PointSize[0.04], 
              Point[{Re[c], Im[c]}]}
}, 
          PlotRange -> {{-1.3, 1.3}, {-1.3, 1.3}}, 
          AspectRatio -> Automatic, Ticks -> None, 
          ImageSize -> {200, Automatic}, 
          Axes -> Automatic]]
]

Arg[]とArcTan[]

?Arg

?ArcTan

ArcTan[1, 
      1]

\(\(\( (*\ 第2象限\ *) \)\(
\)\(ArcTan[\(-1\), 1]\)\)\)

\(\(\( (*\ 第3象限\ *) \)\(
\)\(ArcTan[\(-1\), \(-1\)]\)\)\)

ArcTan[
      1, -1]

ArcTan[0, I]

ArcTan[0, -I]

点Pの極座標表示

(Subscript[r, 1]) = (Sqrt[(2 + I 2) (2 - I 2)])

(Subscript[θ, 1]) = Arg[2 + I 2]

ArcTan[2, 2]

点Qの極座標表示

(Subscript[r, 2]) = (Sqrt[Expand])[(-1 + I (Sqrt[3])) (-1 - I (Sqrt[3]))]

(Subscript[θ, 1]) = Arg[-1 + I (Sqrt[3])]

ArcTan[-1, (Sqrt[3])]

四則演算（加法と減法）

Arg[
      1 + I (2 + (Sqrt[3]))]

Arg[
      3 + I (2 - (Sqrt[3]))]

\(\(\( (*\ 簡約してみる\ *) \)\(
\)\(FullSimplify[
      Arg[3 + I\ \((2 - \@3)\)]]\)\)\)

Arg[
        3 + I (2 - (Sqrt[3]))] // N

四則演算（乗法と除法）

Arg[-2 - 
        2 (Sqrt[ 3]) + I (-2 + 2 (Sqrt[ 3]))]

\(\(\( (*\ 簡約してみる\ *) \)\(
\)\(FullSimplify[
      Arg[\(-2\) - 
          2 \@ 3 + I\ \((\(-2\) + 2 \@ 3)\)]]\)\)\)

\(\(\( (*\ 偏角θ = 
        arg \((\(\(-1\) + \@3\)\/2 + I\ \((\(\(-1\) - \@3\)\/2)\))\)*) \)\(
\)\(Arg[\(\(-1\) + \@3\)\/2 + I\ \((\(\(-1\) - \@3\)\/2)\)]\)\)\)

\(\(\( (*\ 簡約してみる\ *) \)\(
\)\(FullSimplify[
      Arg[\(\(-1\) + \@3\)\/2 + I\ \((\(\(-1\) - \@3\)\/2)\)]]\)\)\)