2003 HSC Question 2 (d) (1 Viewer)

[Neborg67]

This question doesn't seem like it's in english! I might have a shot at the answer if I understand the question.

The question is:

By applying de Moivre's theorem and by also expanding (cos(theta) + i sin(theta))^5, express cos(5theta) as a polynomial in cos(theta).

I have no idea what they want you to do... please help.

 

[KFunk]

For this one you express (cosθ + isinθ )⁵ in two ways and then you equate real parts. I'll show you what I mean:

Using de moivre's (cosθ + isinθ )⁵ = cos5θ + isin5θ


Using binomial expansion (cosθ + isinθ )⁵ becomes:

cos⁵θ + 5isinθcos⁴θ - 10sin²θcos³θ - 10isin³θcos²θ + 5sin⁴θcosθ + isin⁵θ

= cos⁵θ - 10sin²θcos³θ+ 5sin⁴θcosθ + i(5sinθcos⁴θ- 10sin³θcos²θ + sin⁵θ )


You know that these two expressions are equivalent so you can equate the real parts of each side so that:

cos5θ = cos⁵θ - 10sin²θcos³θ+ 5sin⁴θcosθ

You can then use the identity sin²θ = 1 - cos²θ to put it all in terms of cos.


Edit: I forgot all the binomial coefficients . Sorry.

 

[Neborg67]

Thats a bit rough putting binomial expansion in complex numbers. Wouldn't have thought of that. (I get what they mean by " as a polynomial in cos theta now"). Well... i guess if i couldn't remember binomial expansion I could do it the long way lol.

(Thanks for your help again!!!)

 

[Captain pi]

This sort of question is highly likely to occur again.

 

[Trev]

Hi Captain.

 

[Stefano]

For this one you express (cosθ + isinθ )⁵ in two ways and then you equate real parts. I'll show you what I mean:

Using de moivre's (cosθ + isinθ )⁵ = cos5θ + isin5θ


Using binomial expansion (cosθ + isinθ )⁵ becomes:

cos⁵θ + 5isinθcos⁴θ - 10sin²θcos³θ - 10isin³θcos²θ + 5sin⁴θcosθ + isin⁵θ

= cos⁵θ - 10sin²θcos³θ+ 5sin⁴θcosθ + i(5sinθcos⁴θ- 10sin³θcos²θ + sin⁵θ )


You know that these two expressions are equivalent so you can equate the real parts of each side so that:

cos5θ = cos⁵θ - 10sin²θcos³θ+ 5sin⁴θcosθ

You can then use the identity sin²θ = 1 - cos²θ to put it all in terms of cos.


Edit: I forgot all the binomial coefficients . Sorry.

These questions are highly likely to occur, but not likely to be a 5th degree. Probably 3rd or 4th.

 

[Trev]

Says who?...

 

[acmilan]

Isnt this a standard 4 unit question? :S

 

[Neborg67]

Isnt this a standard 4 unit question? :S

I'm not sure. First time i've seen it. I should probably check some more past HSC papers.

 

[Neborg67]

Please tell me ur just starting 4 unit this term...otherwise i'd do a crap load of revision if i were u

umm.... no actually. I'm doing the HSC this year.

 

[Neborg67]

umm.... no actually. I'm doing the HSC this year.

 

[justchillin]

Please tell me ur just starting 4 unit this term...otherwise i'd do a crap load of revision if i were u

 

[Neborg67]

Why did u think I was just starting?

 

[yrtherenonames]

Maybe the wording confused him, although I really don't think it's that obscure.
hehe or maybe the wording's confused us, no-one's actually asked if Neborg's doing 4unit maths for his hsc

 

[Neborg67]

yes yrtherenonames, I am doing 4 unit this year. And Yes, the wording confused me. (and yes, I know i should be up to scratch on complex numbers by now)

 

[Antwan23q]

Please tell me ur just starting 4 unit this term...otherwise i'd do a crap load of revision if i were u

damn man, u always give everyone attitude