A complex number z satisfies the inequality 
Find the least possible value of |z|.
Find the greatest possible value of Arg z.
The inequality defines a circle with center
and radius of 2 units.
Have a look at the diagram. The least possible value of |z| will be the distance from the line connecting the origin to the center of the circle minus the radius of the circle.
So find the magnitude of the origin to the center of the circle, then take away the radius of the circle:
^2+(-2\sqrt{3})^2}=4 \implies |Z|_{min}=4-2=2)
The maximum argument of z will be the argument of the line connecting the origin to the tangent to the circle (purple line) - hopefully the diagram makes that clearer! So you can see we've formed a right angled triangle (using a bit of geometry). We just need to find the angle between the blue line and the horizontal, and the angle between the blue and purple line to find our desired angle.
=\frac{\pi}{6})
(this is the angle between the purple and blue line!)
Now we need to find the angle between the blue line and the x axis (horizontal):
=\frac{\pi}{3})
To angle between the purple line and the horizontal is just the angle between the blue line and horizontal, minus the angle between the blue and purple line:
_{max}=\pi-\frac{\pi}{6}=\frac{5\pi}{6})
Hope that helps! (there may be better ways of doing this... not sure haha)
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