hard complex numbers

Identity · 2009-04-04 04:36:19

Let

. Let . Hint: Use

a) Find the relation between x and y such that:
i)

ii)

b) In each case describe carefully the locus of z.

I don't know how to start, help please!

Last edited by Identity (2009-04-04 04:36:41)

mathsyperson · 2009-04-04 06:08:00

Modulus-argument form seems like the thing to use here.

Set z = r(cosθ + i sinθ).
Then z² = r²(cos(2θ) + i sin(2θ)), and you get a similar thing for z³.

That should let you find conditions on r and θ for the power of z being in S, and then you can get conditions on x and y by using those.

Identity · 2009-04-04 07:24:23

Wait, so is this right for

?

and

If

, and

So

But here

isn't any single value, rather it is within a range of values, how can you change this to x and y?

mathsyperson · 2009-04-04 08:33:40

That looks right, except that θ could also be in [9π/8,7π/6].
You can use the modulus part to say x²+y² = 3, and the arg part will give a restriction as well. I'd guess that's what you're meant to say.

Identity · 2009-04-04 17:03:33

Thanks mathsy

Math Is Fun Forum

#1 2009-04-04 04:36:19

hard complex numbers

#2 2009-04-04 06:08:00

Re: hard complex numbers

#3 2009-04-04 07:24:23

Re: hard complex numbers

#4 2009-04-04 08:33:40

Re: hard complex numbers

#5 2009-04-04 17:03:33

Re: hard complex numbers

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