The area of the triangle whose vertices are represented by `0,z,z e^(ialpha)`

Options

A. `1/2|z|^(2)cosalpha`
B. `1/|z|^(2)sinalpha`
C. `1/2|z|^(2)sinalphacosalpha`
D. `1/2|z|^(2)`

Explanation

To find the area of the triangle whose vertices are represented by the complex numbers \(0\), \(z\), and \(z e^{i\alpha}\), we can follow these steps:

### Step 1: Identify the vertices
The vertices of the triangle are:
- Vertex A: \(0\) (which corresponds to the point \((0, 0)\))
- Vertex B: \(z\) (which can be expressed as \(x + iy\), where \(x\) and \(y\) are the real and imaginary parts of \(z\))
- Vertex C: \(z e^{i\alpha}\) (which can be expressed as \(z (\cos \alpha + i \sin \alpha) = z \cos \alpha + i z \sin \alpha\))

### Step 2: Express the coordinates of the vertices
From the above, we have:
- Vertex A: \((0, 0)\)
- Vertex B: \((x, y)\)
- Vertex C: \((z \cos \alpha, z \sin \alpha)\)

### Step 3: Use the formula for the area of a triangle
The area \(A\) of a triangle given vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the determinant formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates of our vertices:
- \(x_1 = 0\), \(y_1 = 0\)
- \(x_2 = x\), \(y_2 = y\)
- \(x_3 = z \cos \alpha\), \(y_3 = z \sin \alpha\)

The area becomes:
\[ A = \frac{1}{2} \left| 0(y - z \sin \alpha) + x(z \sin \alpha - 0) + (z \cos \alpha)(0 - y) \right| \]
This simplifies to:
\[ A = \frac{1}{2} \left| x z \sin \alpha - z \cos \alpha y \right| \]

### Step 4: Factor out \(z\)
We can factor out \(z\) from the expression:
\[ A = \frac{z}{2} \left| x \sin \alpha - y \cos \alpha \right| \]

### Step 5: Substitute \(z\) in terms of its modulus
Recall that \(z = x + iy\), so:
\[ |z|^2 = x^2 + y^2 \]
Thus, we can express \(x^2 + y^2\) as \(|z|^2\).

### Step 6: Final expression for the area
The area of the triangle can be expressed as:
\[ A = \frac{1}{2} |z|^2 \sin \alpha \]

### Conclusion
The area of the triangle whose vertices are represented by \(0\), \(z\), and \(z e^{i\alpha}\) is:
\[ A = \frac{1}{2} |z|^2 \sin \alpha \]