convert xt 5cos10t30 2sin10t206cos10t80 into a single sinu

convert x(t) = 5cos(10t+30) + 2sin(10t-20)+6cos(10t+80) into a single sinusoid

To convert from cos (omega t - theta) to sin(cot) + cos (omega t), C cos(2 pi ft - theta) = C cos (theta) cos (2 pi ft) + C sin (theta) sin (2 pi ft) = a cos(2 pi ft) + b sin 92 pi ft) where: a = C cos (theta); b = C sin (theta) The magnitude: a^2 + b^2 = C^2 (cos^2 theta + sin^2 theta) = C^2 C = Squareroot a^2 + b^2 The phase: b/a = C sin(theta/C cos(theta) = tan(theta) and... theta = tan^-1(b/a)

Solution

x(t) = 5cos(10t+30) + 2sin(10t-20)+6cos(10t+80)

   = 5cos10t*cos30 - 5*sin10t*sin30 + 2*sin10t*cos20 - 2*cos10t*sin20 + 6cos10t*cos80 - 6*sin10t*sin80

   = 4.33*cos10t - 2.5*sin10t + 1.88*sin10t - 0.68*cos10t + 1.04*cos10t - 5.9*sin10t

   = 1.69cos10t - 6.52*sin10t