Illustrate in the complex plane the following set of points and explain your answer
arg `(Z) = (pi)/(6)`

Answer

To illustrate the set of points in the complex plane where \( \arg(Z) = \frac{\pi}{6} \), we will follow these steps:

### Step 1: Understand the Complex Plane
The complex plane consists of a horizontal axis (the real axis) and a vertical axis (the imaginary axis). Any complex number \( Z \) can be represented as \( Z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

### Step 2: Define the Argument of a Complex Number
The argument of a complex number, denoted as \( \arg(Z) \), is the angle \( \theta \) that the line representing the complex number makes with the positive direction of the real axis. It is measured in radians.

### Step 3: Given Condition
We are given that \( \arg(Z) = \frac{\pi}{6} \). This means that the angle \( \theta \) between the line representing the complex number \( Z \) and the positive real axis is \( \frac{\pi}{6} \) radians.

### Step 4: Determine the Points
To find the points that satisfy this condition, we can use the tangent function, which relates the angle to the ratio of the imaginary part to the real part:
\[ \tan\left(\frac{\pi}{6}\right) = \frac{b}{a} \]
We know that \( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \). Therefore, we can write:
\[ \frac{b}{a} = \frac{1}{\sqrt{3}} \implies b = \frac{a}{\sqrt{3}} \]

### Step 5: Illustrate the Line
The equation \( b = \frac{a}{\sqrt{3}} \) represents a line through the origin with a slope of \( \frac{1}{\sqrt{3}} \). This line will extend infinitely in both directions in the complex plane.

### Step 6: Conclusion
The set of points where \( \arg(Z) = \frac{\pi}{6} \) corresponds to all points along the line that makes an angle of \( \frac{\pi}{6} \) with the positive real axis. This line can be visualized as a straight line passing through the origin, inclined at \( \frac{\pi}{6} \) radians.