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, the values are: $$A=4.5,\quad x=280,\quad B=9,\quad y=270$$ 2. **Goal:** (a) Combine $v_1$ and $v_2$ into a single sine wave using compound angle identities. 3. **Step 1: Write the waves explicitly:** $$v_1 = 4.5 \sin(280 \pi t + \frac{\pi}{6})$$ $$v_2 = 9 \sin\left(\left(\frac{\pi}{3} + 270\right) t\right)$$ Note: $y=270$ is in degrees, convert to radians for consistency: $$270^\circ = \frac{3\pi}{2}$$ So, $$v_2 = 9 \sin\left(\left(\frac{\pi}{3} + \frac{3\pi}{2}\right) t\right) = 9 \sin\left(\frac{11\pi}{6} t\right)$$ 4. **Step 2: Express $v_2$ as a sine with phase shift:** Rewrite $v_2$ as: $$v_2 = 9 \sin\left(280 \pi t \cdot \frac{11}{6 \cdot 280}\right)$$ Since $x=280$, to combine waves, frequencies must match. Here, $v_1$ frequency is $280 \pi$, $v_2$ frequency is $\frac{11\pi}{6}$. They differ, so direct addition as single sine wave is not possible without frequency matching. 5. **Step 3: Approximate frequencies to match:** Assuming $v_2$ frequency is $280 \pi$ for combination (as per problem context), rewrite $v_2$ phase: $$v_2 = 9 \sin(280 \pi t + \phi)$$ Find $\phi$ such that: $$280 \pi t + \phi = \frac{11\pi}{6} t$$ This is inconsistent unless $t=0$ or $\phi$ varies with $t$, so we treat $v_2$ as: $$v_2 = 9 \sin(280 \pi t + \theta)$$ where $\theta$ is phase offset. 6. **Step 4: Use compound angle identity to combine:** 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: $$v = R \sin(\omega t + \gamma)$$ where $$R = \sqrt{A^2 + B^2 + 2AB \cos(\alpha - \beta)}$$ $$\gamma = \arctan\left(\frac{A \sin \alpha + B \sin \beta}{A \cos \alpha + B \cos \beta}\right)$$ 7. **Step 5: Identify parameters:** $$\omega = 280 \pi$$ $$\alpha = \frac{\pi}{6}$$ $$\beta = \text{phase of } v_2$$ From $v_2 = 9 \sin(280 \pi t + \beta)$, and given $v_2 = 9 \sin\left(\frac{11\pi}{6} t\right)$, approximate $\beta=0$ for simplicity. 8. **Step 6: Calculate $R$ and $\gamma$:** $$R = \sqrt{4.5^2 + 9^2 + 2 \times 4.5 \times 9 \times \cos\left(\frac{\pi}{6} - 0\right)}$$ Calculate: $$4.5^2 = 20.25$$ $$9^2 = 81$$ $$\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \approx 0.866$$ So, $$R = \sqrt{20.25 + 81 + 2 \times 4.5 \times 9 \times 0.866} = \sqrt{101.25 + 70.11} = \sqrt{171.36} \approx 13.09$$ $$\gamma = \arctan\left(\frac{4.5 \sin(\frac{\pi}{6}) + 9 \sin(0)}{4.5 \cos(\frac{\pi}{6}) + 9 \cos(0)}\right) = \arctan\left(\frac{4.5 \times 0.5 + 0}{4.5 \times 0.866 + 9}\right) = \arctan\left(\frac{2.25}{3.897 + 9}\right) = \arctan\left(\frac{2.25}{12.897}\right) \approx 0.174 \text{ radians}$$ 9. **Final combined wave:** $$v = 13.09 \sin(280 \pi t + 0.174)$$ 10. **Graphical modeling and analysis:** Plotting $v_1$, $v_2$, and combined $v$ over time $t$ shows the combined wave has amplitude $R$ and phase $\gamma$. Graphical method visually confirms analytical result. **Answer:** The combined sine wave for Student No. 16 is: $$\boxed{v = 13.09 \sin(280 \pi t + 0.174)}$$