周期関数と三角関数

sin θ のグラフ

Plot[Sin[θ], {θ, -Pi, 3  Pi}, 
      AspectRatio -> Automatic, ImageSize -> {400, Automatic}, 
      PlotRange -> {-2, 2}, 
      Ticks -> {{-Pi, -Pi/2, 0, Pi/2, Pi, 3  Pi/2, 2  Pi, 
            5  Pi/2, 3  Pi}, {-1, 0, 1}}]

cos θ のグラフ

Plot[Cos[θ], {θ, -Pi, 3  Pi}, 
      AspectRatio -> Automatic, ImageSize -> {400, Automatic}, 
      PlotRange -> {-2, 2}, 
      Ticks -> {{-Pi, -Pi/2, 0, Pi/2, Pi, 3  Pi/2, 2  Pi, 
            5  Pi/2, 3  Pi}, {-1, 0, 1}}]

tan θ のグラフ

Plot[Tan[θ], {θ, -Pi, 3  Pi}, 
      AspectRatio -> Automatic, ImageSize -> {400, Automatic}, 
      PlotRange -> {-4, 4}, 
      Ticks -> {{-Pi, -Pi/2, 0, Pi/2, Pi, 3  Pi/2, 2  Pi, 
            5  Pi/2, 3  Pi}, {-3, -2, -1, 0, 1, 2, 
            3}}]

sin(θ+(π/2)) とsin θ のグラフ

Plot[{Sin[θ + Pi/2], Sin[θ]}, {θ, -Pi, 3  Pi}, 
      AspectRatio -> Automatic, ImageSize -> {400, Automatic}, 
      PlotRange -> {-2, 
          2}, 
PlotStyle -> {{RGBColor[0, 0, 
              1]}, {RGBColor[1, 0, 0]}}, 
      Ticks -> {{-Pi, -Pi/2, 0, Pi/2, Pi, 3  Pi/2, 2  Pi, 
            5  Pi/2, 3  Pi}, {-1, 0, 1}}]

sin(θ-(π/2)) とsin θ のグラフ

Plot[{Sin[θ - Pi/2], Sin[θ]}, {θ, -Pi, 3  Pi}, 
      AspectRatio -> Automatic, ImageSize -> {400, Automatic}, 
      PlotRange -> {-2, 
          2}, 
PlotStyle -> {{RGBColor[0, 0, 
              1]}, {RGBColor[1, 0, 0]}}, 
      Ticks -> {{-Pi, -Pi/2, 0, Pi/2, Pi, 3  Pi/2, 2  Pi, 
            5  Pi/2, 3  Pi}, {-1, 0, 1}}]

波としての三角関数

円周上を点が回転している様子

\(For[i = 0, i < 2  Pi, 
      i = i + Pi/24, 
Show[
        Graphics[{{RGBColor[0.5, 0.5, 0.5], 
              Circle[{0, 0}, 1]}, {PointSize[0.04], 
              Point[{Cos[i], Sin[i]}]}}], AspectRatio -> Automatic, 
        Axes -> Automatic, ImageSize -> {200, Automatic}, 
        PlotRange -> {{\(-1.3\), 1.3}, {\(-1.3\), 1.3}}, 
        Ticks -> {{{\(-1\), "-A"}, 
              0, {1, "A"}}, {{\(-1\), "-A"}, {0, ""}, {1, "A"}}}]
]\)

回転する点とsin波形 x(t)=A sin(ωt) の関係

\(For[i = 0, i ≤ 4  Pi, 
      i = i + Pi/
            12, 
\(Show[{
Graphics[{RGBColor[0.7, 0.7, 0.7], Circle[{0, 0}, 1]}], 
Graphics[
              Line[{{Pi/2, 
                    1.3}, {Pi/
                      2, \(-1.3\)}}]], 
Graphics[{RGBColor[
                  0.7, 0.7, 0.7], 
                Line[{{Cos[i], Sin[i]}, {Pi/2 + i, Sin[i]}}]}], 
            Graphics[{PointSize[0.01], 
                Point[{Cos[i], 
                    Sin[i]}]}], 
Graphics[{PointSize[0.01], 
                Point[{i + Pi/2, Sin[i]}]}], 
Graphics[
              Plot[Sin[x - Pi/2], {x, Pi/2 - 0.0000001, Pi/2 + i}, 
                DisplayFunction -> Identity]]
}, 
          AspectRatio -> Automatic, Axes -> Automatic, 
          ImageSize -> {800, Automatic}, 
          PlotRange -> {{\(-1.3\), 4  Pi + 2.3}, {\(-1.3\), 1.3}}, 
          Ticks -> {{{\(-1\), "-A"}, {0, ""}, {1, "A"}, {Pi/
                    2, ""}, {Pi, ""}, {3  Pi/2, ""}, {2  
                    Pi, ""}, {5  Pi/2, ""}, {3  Pi, ""}, {7 
                     Pi/2, ""}, {4  Pi, ""}, {9 
                     Pi/2, ""}}, {{\(-1\), "-A"}, {0, ""}, {1, "A"}}}];\)
]\)

振幅の違いによる波形の比較

a = 4;
Plot[{a*Sin[t], Sin[t]}, {t, 0, 13}, AspectRatio -> Automatic, 
      ImageSize -> {400, 
          Automatic}, 
PlotStyle -> {{RGBColor[0, 0, 
              1]}, {RGBColor[1, 0, 0]}}]

角速度の違いによる波形の比較

ω = Pi;
Plot[{Sin[ω t], Sin[2  Pi t]}, {t, 0, 5}, 
      AspectRatio -> Automatic, 
      ImageSize -> {400, 
          Automatic}, 
PlotStyle -> {{RGBColor[0, 0, 
              1]}, {RGBColor[1, 0, 0]}}]

初期位相の違いによる波形の比較

ϕ = Pi/3;
Plot[{Sin[2 Pi t + ϕ], Sin[2  Pi t]}, {t, 0, 5}, 
      AspectRatio -> Automatic, 
      ImageSize -> {400, 
          Automatic}, 
PlotStyle -> {{RGBColor[0, 0, 
              1]}, {RGBColor[1, 0, 0]}}]