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The image displays an electric circuit problem involving alternating current (AC) and direct current (DC) components. The function describing the current \(I(t)\) is given as:

\[
I(t) = I_B \cos(\omega_B t) + I_A \sin(\omega_A t) + I_{DC}
\]

### Provided Information:
- \( I_A = 2 \, \text{A} \)
- \( \omega_A = \frac{1}{2} \, \text{s}^{-1} \)
- \( I_B = 1 \, \text{A} \)
- \( \omega_B = \frac{1}{3} \, \text{s}^{-1} \)
- \( I_{DC} = 1 \, \text{A} \)
- \( t_1 = 9 \, \text{s} \)

The task is to:
1. **Find the algebraic expression for the electric current at \( t = t_1 \).**
2. **Verify the current \( I(t_1) \).**

### Step-by-Step Solution:
1. **Insert values into the equation**:
   We will substitute \( t_1 = 9 \, \text{s} \), \( I_A = 2 \), \( \omega_A = \frac{1}{2} \), \( I_B = 1 \), \( \omega_B = \frac{1}{3} \), and \( I_{DC} = 1 \) into the current equation:

   \[
   I(t_1) = I_B \cos\left(\omega_B t_1\right) + I_A \sin\left(\omega_A t_1\right) + I_{DC}
   \]

2. **Compute \( \cos(\omega_B t_1) \) and \( \sin(\omega_A t_1) \)**:
   - \( \omega_B t_1 = \frac{1}{3} \times 9 = 3 \, \text{radians} \)
   - \( \omega_A t_1 = \frac{1}{2} \times 9 = 4.5 \, \text{radians} \)

   Now calculate the trigonometric values:
   - \( \cos(3) \approx -0.98999 \)
   - \( \sin(4.5) \approx -0.97753 \)

3. **Substitute the values into the current equation**:
   \[
   I(t_1) = 1 \times (-0.98999) + 2 \times (-0.97753) + 1
   \]
   \[
   I(t_1) = -0.98999 - 1.95506 + 1
   \]
   \[
   I(t_1) \approx -1.94505 \, \text{A}
   \]

### Final Answer:
The electric current at \( t_1 = 9 \, \text{s} \) is approximately \( I(t_1) = -1.945 \, \text{A} \).

Would you like further details or clarification? Here are a few follow-up questions:

1. How does the DC component \( I_{DC} \) influence the total current?
2. What is the significance of the angular frequencies \( \omega_A \) and \( \omega_B \) in this circuit?
3. How would the current change if \( t_1 \) were doubled?
4. What is the role of trigonometric functions in describing alternating currents?
5. Can you explain how phase shifts affect AC circuits?

**Tip**: In AC circuit problems, always start by identifying the amplitude, frequency, and phase of each component for easier manipulation of the equations.

The query involves determining the algebraic expression for the electric current at t = t1 and verifying the answer for an AC-DC combined circuit.

Learn how to calculate the electric current in an AC-DC combined circuit at t = t1 using sinusoids and algebra. This problem covers key trigonometric concepts and circuit analysis techniques, suitable for undergraduate electrical engineering or physics students.

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