I have two sinusoidal functions: x1(t)=sin(30t) and x2(t)=0.7sin(30t + 9)
Now I want to get y(t)=x1(t) + x2(t)
I want to get y(t) as a sinusoidal function.
I'm supposed to solve this using Euler's Formula but I can't seem to solve it.
Any help would be greatly appreciated.
Here's what I've tried so far:
http://i.imgur.com/RT7Ghs7.jpg?1
(Please excuse the shameful handwriting)
Best Answer
You can work by assuming that the given signals are the imaginary parts of complex exponentials, and compute
$$e^{i30t}+0.7e^{i(30t+9)}=(1+0.7e^{i9})e^{i30t}.$$
The constant factor is, in polar form,
$$1+0.7e^{i9}=1+0.7\cos(9)+i0.7\sin(9)=\sqrt{(1+0.7\cos(9))^2+(0.7\sin(9))^2}e^{i\arctan(0.7\sin(9)/(1+0.7\cos(9)))},$$
and
$$(1+0.7e^{i9})e^{i30t}=\sqrt{(1+0.7\cos(9))^2+(0.7\sin(9))^2}e^{i(30t+\arctan(0.7\sin(9)/(1+0.7\cos(9))))},$$
giving
$$y(t)=\sqrt{(1+0.7\cos(9))^2+(0.7\sin(9))^2}\sin(30t+\arctan(0.7\sin(9)/(1+0.7\cos(9)))).$$