# Minimum argument of a complex number

1. Dec 28, 2009

### Amaru58

1. The problem statement, all variables and given/known data

Find the minimum value of arg(z) where z satisfies the inequality |z + 3 -2i| </_ 2

2. Relevant equations

Is this working correct? Thank you for help in advance

3. The attempt at a solution

Z lies on a circle with radius 2 and centre -3,2

arg(z)min = pi - 2 tan^-1(2/3)?
= 113 degrees

2. Dec 28, 2009

### futurebird

First note that any argument given by a point inside the circle is also given by a point on the circle so we need only look on the circle for solutions. Then the argument will be smallest when the tangent line to the circle passes through the origin since that will make the angle between the imaginary axis and the point on the circle as small as possible.

3. Dec 28, 2009

### Amaru58

So if we draw two lines: 1) Joining the centre of the circle to the origin and 2) extending a tangent of the circle to the origin as this is where the minimum argument occurs?
Consequently two congruent triangles are formed (if we draw a line down form the centre of the circle perpendicular to the imaginary axis).
We can then find the minimum argument of this complex number as: pi - tan2/3
Therefore the minimum argument is 113 degrees which is 1.97 radians
I'm still not sure what is wrong with my working