ATAR Notes: Forum

HSC Stuff => HSC Maths Stuff => HSC Subjects + Help => HSC Mathematics Extension 2 => Topic started by: Nagisa on December 13, 2012, 10:39:32 pm

Title: polynomials help
Post by: Nagisa on December 13, 2012, 10:39:32 pm: More polynomials coz #YOLO,

Show that $\cos\ 4\theta = 8\cos^{4}\theta - 8\cos^{2}\theta + 1$

a) Hence, solve $8x^{4}-8x^{2}+1=0$ and deduce the exact values of $\cos\ \frac{\pi}{8}$ and $\cos\ \frac{5\pi}{8}$

b) Solve $16x^{4}-16x^{2}+1=0$ and deduce the exact values of $\cos\ \frac{\pi}{12}$ and $\cos\ \frac{5\pi}{12}$

keen to learn some "elegance", is there any short cuts?
Title: Re: Halp pls
Post by: ninwa on December 13, 2012, 10:42:28 pm: Can you make your thread titles more informative in future thanks
Title: Re: Halp pls
Post by: Stick on December 13, 2012, 10:42:29 pm: I've noticed you're from New South Wales. Is your Maths similar to Mathematical Methods (CAS)? This could be done fairly straightforwardly in Specialist Mathematics but I'm hesitant to show you how to do it if it's not part of your study design. That being said, I'm not sure what other way this question can be tackled.
Title: Re: Halp pls
Post by: TrueTears on December 13, 2012, 10:44:34 pm: $\cos(2\theta) = 2\cos^2(\theta)-1$

$\cos(4\theta) = 2\cos^2(2\theta)-1$

$\cos(4\theta) = 2\left[2\cos^2(\theta)-1\right]^2-1 = 8\cos^{4}\theta - 8\cos^{2}\theta + 1$

a) Make the substitution $u = x^2$

$8u^2-8u+1=0$

$u = \frac{\pm \sqrt{2}+2}{4} \implies x = \pm \sqrt{\frac{\pm \sqrt{2}+2}{4}}$

Let $x = \cos(\theta)$

So we get $\cos(4\theta) = 0 \implies \theta = \frac{\pi}{8}+\frac{\pi}{2}k, k \in \mathbb{Z}$

Can you do the rest?
Title: Re: Halp pls
Post by: Nagisa on December 13, 2012, 10:46:29 pm: Quote from: Stick on December 13, 2012, 10:42:29 pm
I've noticed you're from New South Wales. Is your Maths similar to Mathematical Methods (CAS)? This could be done fairly straightforwardly in Specialist Mathematics but I'm hesitant to show you how to do it if it's not part of your study design. That being said, I'm not sure what other way this question can be tackled.

yeah hi, after u gave me hate for no reason i dont like u. that aside, this is an extension 2 questions since i do extension 2, i have solved the question. i just wanted to see other people's methods.
Title: Re: polynomials help
Post by: Stick on December 13, 2012, 10:47:20 pm: I would've tackled it in the same way as TrueTears. Hope that method helps. :)
Title: Re: polynomials help
Post by: Nagisa on December 13, 2012, 10:51:39 pm: Quote from: Stick on December 13, 2012, 10:47:20 pm
I would've tackled it in the same way as TrueTears. Hope that method helps. :)

yeah same, but my teacher pointed out you can use de moivre's theorem to show the first part. seeing if anyone picked up on that also.