In the image you've provided, it appears to show a set of sinusoidal functions and their respective graphs. The exercise seems to involve the superposition (sum) of sinusoidal functions with equal angular frequencies. Here's a breakdown of the components and the key equations shown:

1. First Equation: $i_1(t) = I_1 \sin(\omega t)$

   • $I_1 = 3 \, \text{A}$
   • $\omega = 2 \, \text{rad/s}$ This represents a sine wave with amplitude $I_1 = 3 \, \text{A}$.

2. Second Equation: $i_2(t) = I_2 \cos(\omega t)$

   • $I_2 = 1 \, \text{A}$
   • $\omega = 2 \, \text{rad/s}$ This represents a cosine wave with amplitude $I_2 = 1 \, \text{A}$.

3. Combined Equation (Sum of the two sinusoids): $i(t) = I_1 \sin(\omega t) + I_2 \cos(\omega t) = I_{12} \cos(\omega t + \phi)$ Here, the two sinusoidal functions are added. The result can be expressed as a single sinusoidal function with a phase shift:

   • $I_1 = 3 \, \text{A}$
   • $I_2 = 1 \, \text{A}$
   • $I_{12} = 7$
   • $\phi = \frac{\pi}{6}$

The graphs below each equation visualize the individual sine and cosine functions, and finally, the resultant sinusoidal function that emerges from their sum.

Would you like more details on how the combination of these sinusoidal functions works or any specific calculations?

Here are five related questions to explore:

1. How do you derive the amplitude $I_{12}$ from the two individual amplitudes $I_1$ and $I_2$?
2. What is the significance of the phase shift $\phi$ in the resultant wave?
3. How does the angular frequency $\omega$ influence the behavior of these sinusoidal functions?
4. What happens if the frequencies $\omega$ of the two functions are different?
5. How can this principle of superposition be applied in alternating current (AC) circuit analysis?

Tip: When combining sinusoidal functions, always remember that a sum of sine and cosine functions of the same frequency can always be expressed as a single sinusoidal function with a phase shift.