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The Question:

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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?

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    Yes I am familiar with representation of complex numbers as points in the plane but not in polar coordinates2012-04-28

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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.

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    @panayiotis: Good. In complex variables, there is a great deal of interplay between the algebra and the geometry. In principle one *could* work out the answers without a picture, but that would miss the main point. And about the one more thing, if you think of $z$ as a vector, it would be the angle the vector joining $(1,2)$ to $z$ makes with the positive $x$-axis.2012-04-28