1. **State the problem:** We are given the function $$s = \frac{4}{3\pi} \sin 3t + \frac{4}{5\pi} \cos 5t$$ and want to understand its behavior and graph. 2. **Formula and explanation:** This function is a sum of sinusoidal functions with different frequencies and amplitudes. The general form is $$s = A \sin(\omega t) + B \cos(\nu t)$$ where $A$ and $B$ are amplitudes, and $\omega$, $\nu$ are angular frequencies. 3. **Analyze amplitudes:** The amplitude of the sine term is $$\frac{4}{3\pi} \approx 0.424$$ and the amplitude of the cosine term is $$\frac{4}{5\pi} \approx 0.255$$. 4. **Frequencies:** The sine term oscillates with frequency 3 (angular frequency 3), and the cosine term oscillates with frequency 5. 5. **Behavior:** The function oscillates smoothly as a combination of these two waves, resulting in a complex waveform with varying peaks and troughs. 6. **Graphing:** The graph will show oscillations centered around zero, with maximum amplitude approximately $$0.424 + 0.255 = 0.679$$ and minimum approximately $$-0.679$$. 7. **Summary:** The function $$s = \frac{4}{3\pi} \sin 3t + \frac{4}{5\pi} \cos 5t$$ is a sum of two sinusoidal waves with different frequencies and amplitudes, producing a smooth oscillating curve centered at zero. **Final answer:** The function is a smooth oscillation combining sine and cosine waves with amplitudes $$\frac{4}{3\pi}$$ and $$\frac{4}{5\pi}$$ and frequencies 3 and 5 respectively.