The two waves \(\eta_{1}=6 \cos \left(k x-\omega t+\frac{1}{8} \pi\right)\) and \(\eta_{2}=8 \sin \left(k x-\omega t+\frac{1}{8} \pi\right)\) are traveling on a stretched string. ( \(a\) ) Find the complex representation of these waves. \((b)\) Find the complex wave equivalent to their sum \(\eta_{1}+\eta_{2}\) and the physical (real) wave that it represents. (c) Endeavor to combine the two waves by working only with trigonometric identities in the real domain.

Using Euler's formula, we can convert the trigonometric functions in the given waves into complex exponentials: \(\eta_1 = 6 \cos(kx-\omega t+\frac{1}{8}\pi) = 3e^{i(kx-\omega t+\frac{1}{8}\pi)} + 3e^{-i(kx-\omega t+\frac{1}{8}\pi)}\) \(\eta_2 = 8 \sin(kx-\omega t+\frac{1}{8}\pi) = 4ie^{i(kx-\omega t+\frac{1}{8}\pi)} - 4ie^{-i(kx-\omega t+\frac{1}{8}\pi)}\)

To find the complex wave equivalent to their sum, we add the complex forms of \(\eta_1\) and \(\eta_2\) together: \(\eta_1 + \eta_2 = (3 + 4i)e^{i(kx-\omega t+\frac{1}{8}\pi)} + (3 - 4i)e^{-i(kx-\omega t+\frac{1}{8}\pi)}\)

Now we need to convert this complex wave back to a real wave, using Euler's formula again: \(\eta_1 + \eta_2 = 2\{(3 + 4i)\cos(kx-\omega t+\frac{1}{8}\pi) + (3 - 4i)\cos(-kx+\omega t-\frac{1}{8}\pi)\}\) \(\eta_1 + \eta_2 = 2\{3\cos(kx-\omega t+\frac{1}{8}\pi) + 4\sin(kx-\omega t+\frac{1}{8}\pi)\}\)

In this step, we will attempt to combine the real waves given, using trigonometric identities, and compare the result with our solution from Step 3: \(\eta_1 + \eta_2 = 6 \cos(kx-\omega t+\frac{1}{8}\pi) + 8 \sin(kx-\omega t+\frac{1}{8}\pi)\) Using the angle addition formula \(\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)\) and \(\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)\), we rewrite the sum as: \(\eta_1 + \eta_2 = 2\{3\cos(kx-\omega t+\frac{1}{8}\pi) + 4\sin(kx-\omega t+\frac{1}{8}\pi)\}\) This matches the result in Step 3, hence the answer is consistent. The real wave that represents the sum of the two waves is given by: \(\eta_1 + \eta_2 = 2\{3\cos(kx-\omega t+\frac{1}{8}\pi) + 4\sin(kx-\omega t+\frac{1}{8}\pi)\}\)