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The image contains two questions related to the **Z-transform** of given sequences, which appear to involve sinusoidal and hyperbolic trigonometric functions.

Let's break down each question.

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### 7) The Z-transform of the sequence \( u_n = \sin(2n - 3) \) is:

The Z-transform for a sinusoidal sequence like \( \sin(\omega n + \theta) \) can be calculated using the standard result:

\[
Z\{ \sin(\omega n + \theta) \} = \frac{z \sin \theta - z^2 \sin (\omega + \theta)}{z^2 - 2z \cos(\omega) + 1}, \quad |z| > 1
\]

In this case, we have \( \omega = 2 \) and \( \theta = -3 \). We need to apply the formula, and based on the choices provided, we can evaluate the correct result by calculating the numerator and denominator step by step.

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### 8) The Z-transform of \( \cosh \left( n\frac{\pi}{2} + \theta \right) \):

For hyperbolic functions like \( \cosh(\omega n + \theta) \), the process is similar to sinusoidal functions, but instead of sine and cosine, the hyperbolic counterparts \( \sinh \) and \( \cosh \) are involved. The Z-transform can be given by:

\[
Z\{ \cosh(\omega n + \theta) \} = \frac{z \cosh(\theta) - \cosh(\omega)}{z^2 - 2 z \cosh(\omega) + 1}, \quad |z| > |e^{\omega/2}|
\]

Here, \( \omega = \frac{\pi}{2} \), and we can substitute that value into the general formula to simplify it, yielding one of the provided options.

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Would you like me to calculate the exact Z-transforms for each of these, or do you need help with a specific step in the process? Let me know if you'd like details!

### 5 Follow-up questions:
1. What are the steps to compute the Z-transform of any sinusoidal function?
2. How does the Z-transform handle hyperbolic functions like \( \cosh \) and \( \sinh \)?
3. Why does the region of convergence \( |z| > 1 \) or \( |z| > e^{\pi/2} \) appear in these problems?
4. How do you differentiate between the numerators for sine and cosine Z-transform formulas?
5. What changes when you compute the Z-transform for sequences involving phase shifts?

**Tip:** The Z-transform is especially useful for solving difference equations and analyzing discrete-time systems in signal processing.

The Z-transform of the sequence un = sin(2n - 3) and cosh(nπ/2 + θ)

Learn how to compute the Z-transform for sinusoidal and hyperbolic sequences such as sin(2n - 3) and cosh(nπ/2 + θ). This problem focuses on key discrete mathematics and signal processing concepts.

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