Complex Numbers Help (1 Viewer)

[Web Addict]

Help Please.
a. Determine the roots of z^4+1=0 in cartesian form and plot them on an Argand diagram.
b. Resolve z^4+1 into real quadractic factors.
c. Divide by z^2 to show cos2x=2(cosx-cos45)(cosx-cos135). (x=theta)
I have done parts a and b, but I have trouble with part b.
Thank you.

 

[Amundies]

I've done parts a and b, but I can't use the latex thingy or whatever that people use to make it look all mathy.

But anyway, z^4+1=0. Therefore, z^4=-1. Now, I assume you know how to change that into rcis theta form. So you get z^4=rcistheta. Then the use De Moivre's theorem to find the roots of z. Then after that, you'll notice that z0 has the conjugate z3, while z1 has the conjugate z2. So then you get (z-z0)(z-z1)(z-z2)(z-z3) = 0 where z0, z1, z2 and z3 are roots. You change z3 into z0 conjugate, and z2 into z1 conjugate. Put the conjugates of each other next to each other then expand it into 2 parts (so you have 2 quadratics) and then use quadratic theory from there.

I've tried to keep it broad, probably best for you to do it yourself.

 

[Web Addict]

I have done parts a and b, but I have trouble with part c. Would anyone be able to help me with that please?

 

[Deliriously]

Do you go Ngo and Sons? Just curious.
anyways, i'll give you a hint! factorise the z's out in your answer for part b)

 

[Web Addict]

Do you go Ngo and Sons? Just curious.
anyways, i'll give you a hint! factorise the z's out in your answer for part b)

I do go to Ngo and Sons.
Thanks for the hint. I will try it out now.

 

[Deliriously]

no problem!