ATAR Notes: Forum

HSC Stuff => HSC Maths Stuff => HSC Subjects + Help => HSC Mathematics Extension 2 => Topic started by: rh45_21 on October 24, 2019, 02:46:34 pm

Title: Maths Exam Questions
Post by: rh45_21 on October 24, 2019, 02:46:34 pm: Hey Jake,

Could you please give me a hand with most specifically a) iii) and b) ii). Thank You so Much!!!
Title: Re: Maths Exam Questions
Post by: fun_jirachi on October 24, 2019, 04:22:33 pm: Hey there!

I'm not Jake, but b) ii) was answered here, as was a) iv).

As for a) iii), we consider the sum of the roots found in ii), like so:
$\operatorname{cis} \left(\frac{\pi}{10}\right) + \operatorname{cis} \left(\frac{5\pi}{10}\right) + \operatorname{cis} \left(\frac{9\pi}{10}\right) + \operatorname{cis} \left(-\frac{3\pi}{10}\right) + \operatorname{cis} \left(-\frac{7\pi}{10}\right) = 0 \\ \text{Considering now the imaginary parts of both sides,} \\ \sin \left(\frac{\pi}{10}\right) + \sin \left(\pi-\frac{\pi}{10}\right) + \sin \left(-\frac{3\pi}{10}\right) + \sin \left(\pi+\frac{3\pi}{10}\right) = -1 \\ \text{Using the fact that} \ \sin x = \sin (\pi-x) \ \text{and that} \ \sin (-x) = -\sin x \\ 2\sin \left(\frac{3\pi}{10}\right) - 2\sin \left(\frac{\pi}{10}\right) = 1 \\ \text{Using the result from (i)} \\ 4\left(\cos \left(\frac{\frac{3\pi}{10}+\frac{\pi}{10}}{2}\right) \times \sin \left(\frac{\frac{3\pi}{10}-\frac{\pi}{10}}{2}\right)\right) = 1 \\ \cos \left(\frac{\pi}{5}\right) \sin \left(\frac{\pi}{10}\right) = \frac{1}{4}$

Hope this helps :)