# Complex Numbers Question HELPPPP !!! (1 Viewer)

#### _mysteryatar_

##### New Member
a) Determine the roots of z^4 + 1 = 0 in cartesian form. Plot them on an Argand diagram.
b) Write z^4 + 1 in terms of real quadratic factors/
c) Divide by z^2 to show that cos(2x) = (cos(x) - cos(45)(cos(x) - cos(135))

Need help with part (c) thanks

#### jathu123

##### Active Member
a) Determine the roots of z^4 + 1 = 0 in cartesian form. Plot them on an Argand diagram.
b) Write z^4 + 1 in terms of real quadratic factors/
c) Divide by z^2 to show that cos(2x) = (cos(x) - cos(45)(cos(x) - cos(135))

Need help with part (c) thanks
pretty sure it should be 1/2 cos(2x) for (c)

$\bg_white z^4 + 1 = (z^2+1)^2-2z^2=(z^2-\sqrt{2}z+1)(z^2+\sqrt{2}z+1) \\ Dividing by z^2, we get : \\ z^2 + z^{-2}=(z+z^{-1}-\sqrt{2})(z+z^{-1}+\sqrt{2}) \\ Applying DMT and cancelling outs the sines: \\ 2 \cos{2\theta}=(2 \cos \theta - \sqrt2)(2 \cos \theta +\sqrt2) = 2(\cos \theta - \sqrt{2}/2)\times 2(\cos \theta +\sqrt{2}/2) \\ = 4(\cos \theta - \cos 45)(\cos - \cos 135) \\ Finally divide each side by 4 to get the identity: \\ \\ \frac{1}{2}\cos 2\theta = (\cos \theta - \cos 45)(\cos \theta - \cos 135)$

meant to write x instead of theta rip

Incase you get confused in the 3rd line, I divided the terms of each factor on the RHS by z (which accounts to dividing the whole side by z^2)

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#### _mysteryatar_

##### New Member
thank you, i didnt write part (c) wrong by the way i think you read it wrong because it says: [2](cos(x) + cos(45))(cos(x) + cos(135))

#### jathu123

##### Active Member
thank you, i didnt write part (c) wrong by the way i think you read it wrong because it says: [2](cos(x) + cos(45))(cos(x) + cos(135))
a) Determine the roots of z^4 + 1 = 0 in cartesian form. Plot them on an Argand diagram.
b) Write z^4 + 1 in terms of real quadratic factors/
c) Divide by z^2 to show that cos(2x) = (cos(x) - cos(45)(cos(x) - cos(135))

Need help with part (c) thanks
no worries!
lol idk if im blind or my laptop display is being dodgy

#### peter ringout

##### New Member
Not sure this hangs together properly?
When you say z^4+1=(z^2-root2z+1)(z^2+root2z+1) this is a factorization of a polynomial. It is an identity true for every complex z.

Suddenly in the next line you are assuming that |z|=1?

For example if z=7 then most certainly 49+1/49 is not 2cos(anything)?

Proof as it stands is quite muddy and misleading.

Best to at least state somewhere that you are restricting the identity to the unit circle.

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