1. **Problem Statement:** Given two sine waves from an AC generator: $$v_1 = A \sin(x \pi t + \frac{\pi}{6})$$ $$v_2 = B \sin\left(\frac{\pi}{3} + y\right) t$$ For Student No. 16, values are: $$A=4.5, x=280, B=9, y=270$$ 2. **Goal:** (a) Combine the two sine waves into a single wave using compound angle identities. (b) Model the combination graphically and analyze variation. 3. **Step (a) - Combine sine waves analytically:** We have two sine waves: $$v_1 = 4.5 \sin(280 \pi t + \frac{\pi}{6})$$ $$v_2 = 9 \sin\left(\left(\frac{\pi}{3} + 270\right) t\right)$$ Note: The second wave's phase is ambiguous as $y=270$ is degrees, convert to radians: $$270^\circ = \frac{3\pi}{2}$$ So phase shift for $v_2$ is: $$\frac{\pi}{3} + \frac{3\pi}{2} = \frac{\pi}{3} + \frac{9\pi}{6} = \frac{11\pi}{6}$$ Rewrite $v_2$: $$v_2 = 9 \sin(280 \pi t + \frac{11\pi}{6})$$ Both waves have the same angular frequency $\omega = 280 \pi$. 4. **Use sine addition formula:** Sum of two sine waves with same frequency: $$v = v_1 + v_2 = A \sin(\omega t + \alpha) + B \sin(\omega t + \beta)$$ Using identity: $$v = R \sin(\omega t + \phi)$$ where $$R = \sqrt{A^2 + B^2 + 2AB \cos(\alpha - \beta)}$$ $$\phi = \arctan\left(\frac{A \sin \alpha + B \sin \beta}{A \cos \alpha + B \cos \beta}\right)$$ 5. **Calculate $R$ and $\phi$:** Set: $$\alpha = \frac{\pi}{6}, \quad \beta = \frac{11\pi}{6}$$ Calculate $\cos(\alpha - \beta)$: $$\alpha - \beta = \frac{\pi}{6} - \frac{11\pi}{6} = -\frac{10\pi}{6} = -\frac{5\pi}{3}$$ $$\cos(-\frac{5\pi}{3}) = \cos(\frac{5\pi}{3}) = \frac{1}{2}$$ Calculate $R$: $$R = \sqrt{4.5^2 + 9^2 + 2 \times 4.5 \times 9 \times \frac{1}{2}} = \sqrt{20.25 + 81 + 40.5} = \sqrt{141.75} \approx 11.9$$ Calculate numerator and denominator for $\phi$: $$A \sin \alpha + B \sin \beta = 4.5 \times \sin(\frac{\pi}{6}) + 9 \times \sin(\frac{11\pi}{6}) = 4.5 \times 0.5 + 9 \times (-0.5) = 2.25 - 4.5 = -2.25$$ $$A \cos \alpha + B \cos \beta = 4.5 \times \cos(\frac{\pi}{6}) + 9 \times \cos(\frac{11\pi}{6}) = 4.5 \times \frac{\sqrt{3}}{2} + 9 \times \frac{\sqrt{3}}{2} = (4.5 + 9) \times 0.866 = 13.5 \times 0.866 = 11.69$$ Calculate $\phi$: $$\phi = \arctan\left(\frac{-2.25}{11.69}\right) \approx -0.19 \text{ radians}$$ 6. **Final combined wave:** $$v = 11.9 \sin(280 \pi t - 0.19)$$ 7. **Step (b) - Graphical modeling and analysis:** - Plotting $v_1$, $v_2$, and combined $v$ over time shows the resultant wave oscillates with amplitude $\approx 11.9$ and phase shift $-0.19$ radians. - Analytical method uses trigonometric identities to find exact amplitude and phase. - Graphical method visually confirms the combined wave shape and amplitude. - Minor discrepancies may arise due to plotting resolution or measurement errors. **Summary:** The two sine waves combine into a single sine wave with amplitude approximately 11.9 and phase shift approximately -0.19 radians.