https://file.aiquickdraw.com/mathbot/file/1722768115099o4lfwzaf.jpg

The equation in the image can be interpreted and solved as follows:

\[
\frac{1}{\infty} + \frac{1}{\infty} = \frac{\text{w}}{2\pi} \cdot \cos(\text{w}t - \theta) \quad \text{as} \quad t \to \infty
\]

### Interpretation:
1. **\(\frac{1}{\infty}\) and \( \frac{1}{\infty}\):**
   - Both expressions \(\frac{1}{\infty}\) represent values that approach 0, since any finite number divided by infinity tends to 0.

2. **Cosine Function:**
   - The equation is trying to solve the equality where \( \frac{\text{w}}{2\pi} \cdot \cos(\text{w}t - \theta) \) as \( t \to \infty \).

### Simplification:
Given that:
\[
\frac{1}{\infty} + \frac{1}{\infty} = 0 + 0 = 0
\]
The equation simplifies to:
\[
0 = \frac{\text{w}}{2\pi} \cdot \cos(\text{w}t - \theta)
\]

For the equality to hold as \( t \to \infty \), the term \( \cos(\text{w}t - \theta) \) must approach 0, meaning that:
\[
\cos(\text{w}t - \theta) = 0
\]

### Solution:
The cosine function equals zero when its argument is an odd multiple of \( \frac{\pi}{2} \), so:
\[
\text{w}t - \theta = (2n + 1)\frac{\pi}{2} \quad \text{where} \quad n \in \mathbb{Z}
\]

Thus, the solution in terms of \( t \) is:
\[
t = \frac{(2n + 1)\frac{\pi}{2} + \theta}{\text{w}} \quad \text{for integers} \quad n
\]

### Summary:
The solution indicates that at specific points in time \( t \), given by the expression above, the equation will hold true as \( t \to \infty \).

Do you have any questions or need further details?

### Related Questions:
1. What are the properties of the cosine function that determine its zeros?
2. How does the concept of a limit apply to \( \frac{1}{\infty} \)?
3. What are the implications of \( t \to \infty \) in real-world scenarios?
4. Can the angular frequency \( \text{w} \) affect the outcome of this equation?
5. How does the phase shift \( \theta \) impact the solution?
6. What is the significance of the expression \( (2n + 1)\frac{\pi}{2} \)?
7. How can you verify the solution for specific values of \( n \)?
8. What would happen if the frequency \( \text{w} \) were zero?

**Tip:** When dealing with infinity in equations, it’s important to remember that any finite number divided by infinity tends to zero.

Explore the solution to the equation involving limits and the cosine function as t approaches infinity. Understand how the cosine function zeros affect the equation's outcome. Suitable for college-level mathematics enthusiasts.

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