𝑧 =
𝑎
𝑠𝑒𝑛 𝛼−𝑖 𝑠𝑒𝑛 𝛼
, (31) 𝑎 ∈ ℝ, 0 ≤ 𝛼 ≤ 2�

The expression given is:

\[
z = a \sin \alpha - i \sin \alpha
\]

where \( a \in \mathbb{R} \) and \( 0 \leq \alpha \leq 2\pi \). Here’s how we can interpret and simplify the expression.

### Step 1: Recognize the structure
The variable \( z \) represents a complex number since it includes the imaginary unit \( i \). Specifically, the expression can be split into a real part and an imaginary part.

\[
z = (a \sin \alpha) + (- \sin \alpha) i
\]

Thus, the real part is \( a \sin \alpha \) and the imaginary part is \( -\sin \alpha \).

### Step 2: Write \( z \) in standard complex form
We now rewrite \( z \) as a standard complex number:

\[
z = (a \sin \alpha) + (- \sin \alpha) i
\]

or more compactly:

\[
z = \sin \alpha (a - i)
\]

This expression shows that the complex number \( z \) is a scaled version of the complex constant \( a - i \), where the scale factor is \( \sin \alpha \).

### Conclusion:
The complex number \( z \) depends on both \( \alpha \) and the constant \( a \), and is expressed as \( \sin \alpha (a - i) \).

Would you like further details on this, or have any specific questions?

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Here are 5 related questions to expand on this information:
1. How does the magnitude of \( z \) change as \( \alpha \) varies from 0 to \( 2\pi \)?
2. What is the geometric interpretation of \( z \) on the complex plane?
3. How does the argument (angle) of \( z \) behave for different values of \( \alpha \)?
4. How does the value of \( a \) affect the real part of \( z \)?
5. Can you express \( z \) in polar form?

**Tip:** When dealing with complex numbers, always think of them as points on the complex plane, where you can analyze both their magnitude and direction (angle).

𝑧 = 𝑎 sin 𝛼 − i sin 𝛼, (31) 𝑎 ∈ ℝ, 0 ≤ 𝛼 ≤ 2π

Understand how to simplify and interpret the complex number expression z = a sin(α) - i sin(α). This solution involves trigonometry and complex numbers, covering the decomposition of the real and imaginary parts. Suitable for undergraduate students.

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