Complex Numbers finding values

[lunds002]

Homework Statement

Given that arg((z₁z₂)/2i) = \pi, find a value of k.

Homework Equations

arg z₂=2\pi=0

arg z₁=\pi/6

The Attempt at a Solution

((\pi/6)^k x (2\pi))/ (2i) = \pi

I'm not sure what to do with the imaginary number i..

 

[gb7nash]

Given that arg((z₁z₂)/2i) = \pi, find a value of k.

You're missing a k in the problem. Where is it missing?

 

[lunds002]

Oops, should be arg ( z1 x z2^k) / (2i)

 

[rock.freak667]

Oops, should be arg ( z1 x z2^k) / (2i)

You should be able to get it by expanding the arg function.

E.g. arg(z₁z₁)=arg(z₁)+arg(z₂).

(the rules are similar to log rules)

 

[lunds002]

Okay so then I get

arg(z1^k) + arg(z2) = pi
2i

(pi/6)^k + 2pi = pi
2i

(pi/6)^k + 2pi = 2i(pi)

Not sure what to do with the imaginary i

 

[rock.freak667]

Okay so then I get

arg(z1^k) + arg(z2) = pi
2i

(pi/6)^k + 2pi = pi
2i

(pi/6)^k + 2pi = 2i(pi)

Not sure what to do with the imaginary i

Check the wikipedia page on argument for how to deal with arg(z₁/z₂). It's similar to how you expand log_a(x/y).

You should know what arg(i) is equal to.

 

[lunds002]

Okay so I know z1/z2 = r1/r2 cis (theta-\psi)

I don't really understand how that applies here though

And you're right, I do know that arg(i) = -1

 

[rock.freak667]

Okay so I know z1/z2 = r1/r2 cis (theta-\psi)

I don't really understand how that applies here though

And you're right, I do know that arg(i) = -1

http://en.wikipedia.org/wiki/Argument_(complex_analysis)#Identities

arg(i) is the angle formed by the z=i and the positive real axis. z=i is the line perpendicular to the positive real axis, so arg(i) is not -1 but ?