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HSC Stuff => HSC Maths Stuff => HSC Subjects + Help => HSC Mathematics Extension 2 => Topic started by: Jimbo123 on November 14, 2019, 09:09:56 pm

Title: Help with complex question
Post by: Jimbo123 on November 14, 2019, 09:09:56 pm: Hi all,

Need some help with the attached question.
Title: Re: Help with complex question
Post by: DrDusk on November 14, 2019, 09:37:05 pm: Quote from: Jimbo123 on November 14, 2019, 09:09:56 pm
Hi all,

Need some help with the attached question.
Could you please share you attempt =) That way were not just doing it all for you but rather solidifying what you know and don't know.

If you've tried and just don't know where to begin, that's alright we can lead you in the right direction.
Title: Re: Help with complex question
Post by: DrDusk on November 14, 2019, 10:02:49 pm: One thing I will do Part (a) fully just because when I did 4u my tutor highlighted something which I think is very important.
Okay so start with:

$S = 1 + z^2 + z^4+...+z^{2n-2} \\ \text{Now using the formula of a Geometric series we get:} \\ S = \dfrac{z^{2n} - 1}{z^2 - 1} \\ \text{Now this is important, you must manually pull out a factor of}\hspace{2mm}z^{n-1}\hspace{2mm}\text{Using index laws} \\ \therefore S = z^{n-1}\bigg(\dfrac{z^{n+1} - z^{1-n}}{z^2-1}\bigg) \\ = z^{n-1}\bigg(\dfrac{z^{n+1} - z^{1-n}}{z^2-1}\times \dfrac{1/z}{1/z}\bigg)\hspace{2mm}\text{As}\hspace{2mm}\dfrac{1/z}{1/z} = 1 \\ = z^{n-1}\bigg(\dfrac{z^n - z^{-n}}{z-z^{-1}}\bigg)$

The 'pulling out a factor' 'trick' is really important because it allows you to manipulate whatever expression you have into looking like the one your trying to prove, because in the end if you've done it all correctly it should reduce to what they've given.
Title: Re: Help with complex question
Post by: david.wang28 on November 15, 2019, 09:35:04 pm: For part ii), Let z^n = cosθ+isinθ (1) and z^(-n) = cosθ-isinθ (2).
(1) - (2), get 2isinθ.
Title: Re: Help with complex question
Post by: david.wang28 on November 16, 2019, 02:48:27 pm: Also for part iii), I have uploaded my solution, let me know if you have any questions (I can't use LaTeX to save my life) :(