Method 1: Graphing

First notice that we are not asked to find \omega. Indeed, regardless of what \omega is, V and \theta will not change. To make our lives easier, we can therefore just take \omega=1.

This problem can be solved by writing the waves as the real part of complex exponentials, but it is probably easier to just graph the sum 3\sin\left(X\right)+8\sin\left(X+120\degree\right) . Make sure your calculator is in degrees!

We immediately notice a new wave, which tells us we're on the right track. To find V, the new amplitude, we simply find the height of the maxima , since the new wave is centered on the x-axis:

V=7

Now consider that w'ere looking for \theta in 7\sin\left(X+\theta\right). This +\theta represents a horizontal translation \theta units to the left. That means every root is shifted left by \theta! The regular \sin function has a root at x=0, from where it goes up (positive).

On our calculator screen, we want to see how far to the left it has been shifted. We look for the first root to the left of x=0 where the function goes from negative to positive . We see that this is at -98.2, which corresponds to a left-wards shift of \theta=98.2\degree.

Answer:

\boxed{V=7,\theta=98.2\degree}

Method 2: Complex phasors

Let's start by writing the waves as the imaginary parts of complex phasors:

3\sin\left(\omega t\right)=\operatorname{\mathrm{Im}}\left(3e^{i\omega t}\right)

8\sin\left(\omega t+120\degree\right)=\operatorname{\mathrm{Im}}\left(8e^{i\left(\omega t+2\pi\/3\right)}\right)

Their sum will therefore be

\begin{align}V_1+V_2 & =\operatorname{\mathrm{Im}}\left(3e^{i\omega t}+8e^{i\omega t+2i\pi\/3}\right)\\ \placeholder{} & =\operatorname{\mathrm{Im}}\left\lbrack e^{i\omega t}\cdot\left(3+8e^{2\pi i\/3}\right)\right\rbrack\end{align}

Using a calculator, we evaluate

3+8e^{2\pi i\/3}=7e^{1.714i}

So the voltage becomes

\begin{align}V_1+V_2 & =\operatorname{\mathrm{Im}}\left(e^{i\omega t}\cdot7e^{1.714i}\right)\\ \placeholder{} & =7\operatorname{\mathrm{Im}}\left(e^{\left(\omega t+1.714\right)i}\right)\\ \placeholder{} & =7\sin\left(\omega t+1.714\right)\end{align}

We once again find the amplitude is V=7, and by converting into radians we find \theta=\frac{180\degree}{\pi}\cdot1.714=98.2\degree!

Complex Phasors

Complex Phasors

Complex Phasors

Complex Phasors