Question:
The point $P$ represents a complex number $z$ in an Argand diagram.
Given that $|z+2-2 \sqrt{3} i|=2$a sketch the locus of $P$ on an
Argand diagram.b Write down the minimum value of $\arg(z)$.
c Find the
maximum value of $\arg(z)$.
I have some gaps in this chapter, and I would like some clarifications. What does arg(z) represent and what does $${\displaystyle \arg \left( z+2-2\, \sqrt{3}i \right) }$$ represent? (the angle from which point to which point)
Also how can I find the minimum and maximum angle?
Best Answer
Hint: Rewrite as $$\left|z-(-2+2\sqrt{3}i)\right|=2.$$ This "says" that the distance from $z$ to $-2+2\sqrt{3}i$ is $2$. So the locus is the circle with a certain centre, a certain radius.
Draw that circle (crucial). You are interested in lines from the origin to points on your circle. The maximum, minimum angles are at points of tangency. One of them will be obvious from the picture. You can work out the other using once familiar geometry/trigonometry. Note that it is $\text{arg}(z)$ that you are being asked about.