Q. Let `z_1`, `z_2,` `z_3` be three vertices of an equilateral triangle circumscribing the circle `|z|=1/2` ,if `z_1=1/2+sqrt3i/2` and `z_1`,`z_2`,`z_3` are in anticlockwise sense then `z_2` is

Options

A. `1+isqrt(3)`
B. `1-isqrt(3)`
C. 1
D. `-1`

Answer

To find the value of \( z_2 \) given \( z_1 = \frac{1}{2} + \frac{\sqrt{3}}{2} i \) and knowing that \( z_1, z_2, z_3 \) are vertices of an equilateral triangle in an anticlockwise direction, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given point \( z_1 \)**:
\[ z_1 = \frac{1}{2} + \frac{\sqrt{3}}{2} i \]
This point can be represented in polar form as:
\[ z_1 = \cos\left(\frac{\pi}{3}\right) + i \sin\left(\frac{\pi}{3}\right) \]
where the modulus \( r = 1 \) (since \( \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = 1 \)) and the argument \( \theta = \frac{\pi}{3} \).

2. **Determine the angle for \( z_2 \)**:
Since \( z_2 \) is the next vertex of the equilateral triangle, we need to rotate \( z_1 \) by \( 120^\circ \) (or \( \frac{2\pi}{3} \) radians) in the anticlockwise direction. The new angle for \( z_2 \) will be:
\[ \theta_2 = \theta_1 + \frac{2\pi}{3} = \frac{\pi}{3} + \frac{2\pi}{3} = \pi \]

3. **Calculate \( z_2 \)**:
Using the polar form:
\[ z_2 = \cos(\pi) + i \sin(\pi) \]
Evaluating this gives:
\[ z_2 = -1 + 0i = -1 \]

4. **Conclusion**:
Therefore, the value of \( z_2 \) is:
\[ z_2 = -1 \]