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I'm pretty sure your proofs are all correct, however you might be able to shorten some of them: 1. In part i) I think you only really needed the first and last steps. 2. In part ii) I don't think you need to write 'are equal', at the end of the angles subtended on the same arc theorem since...

Carrotsticks, do you have solutions to these papers? They would make decent supplementary practice for everyone

Re: HSC 2016 Complex Numbers Marathon part iii) please http://i.imgur.com/sP2ojE1.png

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread part b... I think... I'm not sure that you can just assume the altitudes meet at E http://i.imgur.com/9qgFLtr.jpg

Re: HSC 2016 4U Marathon thanks, but what about algebraically? can you please give me a hint or two? I started with by squaring both sides, then writing them as the products complex numbers and conjugates but im not sure if thats on the right track

Re: HSC 2016 Complex Numbers Marathon http://i.imgur.com/VOCNtil.png

Re: HSC 2016 4U Marathon http://i.imgur.com/VOCNtil.png

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread By the way, thanks for making this thread David

Re: HSC 2016 3U Marathon nvm, a)\quad { (\sin { x } +\cos { x } ) }^{ 2 }\\ =\sin ^{ 2 }{ x } +\cos ^{ 2 }{ x } +2\sin { x } \cos { x } \\ =1+2\sin { x } \cos { x } \\ b)\quad \int { \frac { \cos { x } -\sin { x } }{ 1+\sin { x } \cos { x } } } dx\\ =\int { \frac { 2(\cos { x } -\sin { x } )...

Re: HSC 2016 3U Marathon is it meant to be 1+2sinxcosx in the denominator?

$I\quad am\quad testing\quad my\quad new\quad cannon.\quad \\ I\quad fire\quad the\quad cannonball\quad upward\quad off\quad a\quad 135m\quad tall\quad cliff\quad and\quad it\quad lands\quad 235m\quad from\quad the\quad base\quad of\quad the\quad cliff.\\ If\quad the\quad cannonball\quad...

Re: HSC 2016 4U Marathon Yep thanks, that's how we were taught to do it but my teacher also showed the cis theta way though he wasnt sure why we could just assume the modulus is 1 (now i know you cant)

Re: HSC 2016 4U Marathon Just a quick question - when solving polynomials with symmetrical co-efficients such as 2x^4 +3x^3 +5x^2 +3x+2=0, if I were to divide by x^2 and use the substitution x=cis theta , how would I know that the roots definitely have modulus 1?

Re: HSC 2016 3U Marathon dw got it now

Re: HSC 2016 3U Marathon 8(i)\quad $Simplify$\quad sin(A+B)-sin(A-B)\\ (ii)\quad $Hence\quad prove\quad by\quad mathematical\quad induction\quad for\quad all\quad positive\quad integers\quad n\quad that$\\ \sin { \theta } +\sin { 2\theta } +\sin { 3\theta } +...+\sin { n\theta } \frac {...

Re: HSC 2016 3U Marathon Paradoxica said there was a way to do it with complimentary angles though, how would you do that?

Re: HSC 2016 3U Marathon I still have no idea how to do this without sums and products - could someone please explain?

Re: HSC 2016 4U Marathon okay so { u }^{ 8 }=1,\quad $where$\quad u=\frac { 1 }{ z } +1,u\neq 1\\ let\quad u=cis\theta \\ cis8\theta =cis0\\ u=cis(\frac { n\pi }{ 4 } ),\quad where\quad n=\pm 1,\pm 2,\pm 3,4\\ 1+\frac { 1 }{ z } =cis(\frac { n\pi }{ 4 } )\\ Now what? Sorry, I am bad

Re: HSC 2016 4U Marathon Hey guys, can you please help me with this? Please don't use Euler's formula, this was in the DeMoivre's chapter