<< Graphics`Graphics`
dB[z_] = 
        20 Log[10, Abs[z]];     
dg[z_] = 
        Im[Log[z]]/Degree ;

example 6.12   p.116

grω = k ((s + 1)/(α s + 1))^2 2/(s + 
                  2) /. s ->  I ω /. {α ->  
            0.1, k ->  1}

LogLinearPlot[{dB[grω], dg[grω] 15/90 + 10}, {ω, 
        0.1, 50}, GridLines ->  Automatic, 
      PlotRange -> {{0.1, 50},  {-5, 25}}, 
      PlotPoints ->  1000]

Q(c)

g1 = dB[grω /. ω ->  2]
θ1 = 
      dg[grω /. ω ->  2]
a1 = 10^(g1/20)

Plot[{0.6 Sin[2 t + 15  °], 
        0.6 a1  Sin[2 t + 15  ° + θ1 °]}, {t, 0, 
        3 Pi/2}, GridLines ->  Automatic, 
      PlotStyle ->  {Hue[0.9], Hue[0.1]}]

Q(d)

g2 = dB[grω /. ω ->  30]
θ2 = 
      dg[grω /. ω ->  30]
a2 = 10^(g2/20)

Plot[{0.9 Sin[30 t + 30 °], 
        0.9 a2  Sin[30 t + 30  ° + θ2 °]}, {t, 0, 
        3 Pi/30}, GridLines ->  Automatic, 
      PlotStyle ->  {Hue[0.9], Hue[0.1]}]

Q(b)

g3 = dB[grω /. ω ->  8]
θ3 = 
      dg[grω /. ω ->  8]
a3 = 10^(g3/20)

Plot[{1 Sin[8 t + 30 °], 
        1 a3  Sin[8 t + 30  ° + θ3 °]}, {t, 0, 
        3 Pi/8}, GridLines ->  Automatic, 
      PlotStyle ->  {Hue[0.9], Hue[0.1]}]

Q(a)

g4 = dB[grω /. ω ->  1]
θ4 = 
      dg[grω /. ω ->  1]
a4 = 10^(g4/20)

Plot[{1 Sin[1 t + 30 °], 
        1 a4  Sin[1 t + 30  ° + θ4 °]}, {t, 0, 
        6 Pi/2}, GridLines ->  Automatic, 
      PlotStyle ->  {Hue[0.9], Hue[0.1]}]