Application of Complex Number

Let X = R1 = 10 cos(wt + h) and Y = R2 = 8cos(wt + k).
Magnitude of X = 10, argument (phase shift) of X = h, so X can be written as 10 cos h + i 10 sin h.
Similarly, Y can be written as 8 cos k + i 8 sin k.
By principle of complex number and Pythagoras theorem, (magnitude of X + Y)^2 = A^2 = (10 cos h + 8 cos k)^2 + (10 sin h + 8 cos k)^2.
And argument of X + Y = arc tan [(10 sin h + 8 sin k)/(10 cos h + 8 cos k)]
(b) Substituting the figure h = 30 degree and k = 60 degree, you can find the value of A and argument ( phase shift) of X + Y. The graph is also sinusoidal with magnitude A and phase shifted.