Complex numbers: deduce that cos 2A = 2(cos A - cos (pi/4))(cos A - cos(3pi/4))

[Harry_the_cat]

Write z⁴+1 in term of its real and quadratic factors

and deduce that

cos 2A = 2(cos A - cos (pi/4))(cos A - cos(3pi/4)).

One of my students has brought this problem to me and I just can't see the light.

Can someone please explain what it means by "real and quadratic" factors? Then I might be able to see the link and do the deduction.

Thanks in advance.

(PS. I know the statement can be proven in various other ways, but I need to follow the question as written.)

 

[mmm4444bot]

Write z⁴+1 in term of its real and quadratic factors

...

Can someone please explain what it means by "real and quadratic" factors?

Perhaps it means quadratic polynomials in z, with Real coefficients.

(z^2 + sqrt[2]*z + 1)(z^2 - sqrt[2]*z + 1) = z^4 + 1 :cool:

 

[Deleted member 4993]

To paraphrase Euler,

Every polynomial with real coefficients, can be factorized as quadratic functions with real coefficient.

 

[Harry_the_cat]

Perhaps it means quadratic polynomials in z, with Real coefficients.

(z^2 + sqrt[2]*z + 1)(z^2 - sqrt[2]*z + 1) = z^4 + 1 :cool:

Yes thankyou.

I still can't see how to do the deduction. I know that cis (pi/4) and cis(3pi/4) are two roots, so that's where those angles come into it, but can't see how to deduce the statement required. Any help appreciated.

 

[Harry_the_cat]

To paraphrase Euler,

Every polynomial with real coefficients, can be factorized as quadratic functions with real coefficient.

Thankyou.

 

[Harry_the_cat]

OK so

\(\displaystyle z^4+1 = (z^2+\sqrt{2}z+1)(z^2-\sqrt{2}z+1)\).

So now I need to deduce the statement: cos 2A = 2(cos A - cos (pi/4))(cos A - cos(3pi/4)).

I'm really stuck as to how to use the factors above to do the deduction.

Please help or is it just a stupid textbook question??