Question

2. Use the algebraic method to generate a summation of the following waves. Note that one of the waves is in sin and would need to be phase shifted into a cos one: E1=7 cos⁡(π/3-ωt), E2=12 sin⁡(π/4-ωt), E3=20 cos⁡(π/5-ωt)

Answer 1

given

E1= 7 cos⁡(π/3-ωt)

E2=12 sin⁡(π/4-ωt),

E3=20 cos⁡(π/5-ωt)

from the trignometric relations  

cos (A-B) = cosA cosB+ sinA*sin B

sin (A-B) = Sin A cos B - cosA sin B

E1 = 7( cos(pi/3) cos (wt) + sin(pi/3) sin (wt))

E2 = 12(sin(pi/4) cos(wt) - cos(pi/4) sin(wt))

E3 = 20( cos(pi/5) cos (wt) + sin(pi/5) sin (wt))

summation of the waves is  

E = E1+E2+E3

E = 7*0.5 cos(wt)+7*0.866 sin (wt) + 12*0.707 cos (wt) - 12*0.707 sin(wt) + 20*0.809 cos(wt)+20*0.59 sin(wt)

E = 3.5 cos(wt) + 6.062 sin (wt) + 8.484 cos(wt)- 8.484 sin(wt) + 16.18 cos(wt) +11.8 sin(wt)

E = 28.164 cos (wt) +9.378 sin (wt)

E = 3.0 cos (wt) + sin (wt)