Complex Algebra: Trigonometry (tan)

[PedroB]

Homework Statement

Let tan(q) with q ε ℂ be defined as the natural extension of tan(x) for real values

Find all the values in the complex plane for which |tan(q)| = ∞

Homework Equations

Expressing tan(q) as complex exponentials:

(e^iq - e^(-iq))/i(e^iq + e^(-iq))

The Attempt at a Solution

I really have no idea how to get around this problem. No matter what I equate 'q' to I don't seem to get a valid answer. Any help would be greatly appreciated.

 

[Dick]

Homework Statement

Let tan(q) with q ε ℂ be defined as the natural extension of tan(x) for real values

Find all the values in the complex plane for which |tan(q)| = ∞

Homework Equations

Expressing tan(q) as complex exponentials:

(e^iq - e^(-iq))/i(e^iq + e^(-iq))

The Attempt at a Solution

I really have no idea how to get around this problem. No matter what I equate 'q' to I don't seem to get a valid answer. Any help would be greatly appreciated.

tan(q) having a pole means the denominator must vanish. So you want to find values of q where e^(iq)+e^(-iq)=0. Multiply both sides by e^(iq).

 

[PedroB]

Ok, so I get q= ∏/2 (which makes sense since tan(∏/2)= ∞), but what does this mean exactly? Are the range of values when x + iy = ∏/2? Surely this simply leads to x = ∏/2 which is evidently not a 'range' of values. This is the simpler of problems that I need to do, though it's the one that's given me more trouble (I've got a few complex derivatives questions which were pretty straight forward to me compared to this, strangely enough)

 

[Dick]

Ok, so I get q= ∏/2 (which makes sense since tan(∏/2)= ∞), but what does this mean exactly? Are the range of values when x + iy = ∏/2? Surely this simply leads to x = ∏/2 which is evidently not a 'range' of values. This is the simpler of problems that I need to do, though it's the one that's given me more trouble (I've got a few complex derivatives questions which were pretty straight forward to me compared to this, strangely enough)

e^(x)=e^(x+2*i*pi*n) for any integer n. So, yes, there are lots of solutions for q.