Complex Numbers

dmg24

It's dmg's quiz hour again ...

I missed my lesson on complex numbers (am off sick at the mo). Can someone help me with this please?

Use the expansion of (cos a + i sin a)^3 to express cos 3a in terms of cos a, and sin 3a in terms of sin a.

Easter Eggs (out of date, from Aldi) are on their way to the top answers! x

pinginal

Twenty-three

jamiex

Have you learned de Moivre's theorem?

I will use 'j' instead of 'i' just because it's easier to see. You can use either letter to represent the square root of -1.

(cos(x)+jsin(x))^n = cos(nx)+jsin(nx)

So in your case, (cos(x)+jsin(x))^3 = cos(3x)+jsin(3x)

Ok, so here it goes:

cos(3x)+jsin(3x) = (cos(x)+jsin(x))^3
Expanding (cos(x)+jsin(x))^3
gives cos^3(x) + 3jcos^2(x)sin(x) - 3cos(x)sin^2(x) - jsin^3(x)

So:

cos(3x)+jsin(3x) = cos^3(x) + 3jcos^2(x)sin(x) - 3cos(x)sin^2(x) - jsin^3(x)

Now, equating the real parts:

cos(3x) = cos^3(x) - 3cos(x)sin^2(x)

But sin^2(x) = (1 - cos^2(x)) because sin^2(x) + cos^2(x) = 1

So
cos(3x) = cos^3(x) - 3cos(x)(1 - cos^2(x))
cos(3x) = cos^3(x) - 3cos(x) + 3cos(x)cos^2(x)
cos(3x) = cos^3(x) - 3cos(x) + 3cos^3(x)

cos(3x) = 4cos^3(x) - 3cos(x)


I hope this has helped. If you have any questions just ask.

Regards,
Jamie

dmg24

Big big thank you (to both of you?!!), I'd got my expansion wrong, so everything after came out wrong too!

Unfortunately I have eaten the chocolate in the meantime ... can offer my mum's very nice Sunday dinner instead?! x

Next topic ... Decision ... be back in five! x

tr3mor

jamiex is right, apart from his pesky engineers' notation.

j instead of i, crazy! :rolleyes:

retepetsir

dmg24 wrote: »

Big big thank you (to both of you?!!), I'd got my expansion wrong, so everything after came out wrong too!

Unfortunately I have eaten the chocolate in the meantime ... can offer my mum's very nice Sunday dinner instead?! x

Next topic ... Decision ... be back in five! x

It doesnt sound like you had the chocolate on offer in the first place

dmg24

retepetsir wrote: »

It doesnt sound like you had the chocolate on offer in the first place

Honestly, there was ... but it was just too tempting ... and am suffering for it now! x

jamiex

dmg24 wrote: »

Big big thank you (to both of you?!!), I'd got my expansion wrong, so everything after came out wrong too!

Unfortunately I have eaten the chocolate in the meantime ... can offer my mum's very nice Sunday dinner instead?! x

Next topic ... Decision ... be back in five! x

You're welcome...and thanks for the offer of Sunday dinner but I'm vegetarian so i'll have to turn you down.

tr3mor wrote: »

jamiex is right, apart from his pesky engineers' notation.

j instead of i, crazy! :rolleyes:

Haven't some exam boards changed their syllabus to use j's instead of i's now? When I did A-level I took my exams with OCR, and i'm pretty sure they did that.

Are you doing Maths at uni by any chance?

emma396

OMG!! i finished my maths degree about 2 years ago. good ol' complex numbers. i^2 = -1

dmg24

jamiex wrote: »

You're welcome...and thanks for the offer of Sunday dinner but I'm vegetarian so i'll have to turn you down.

Haven't some exam boards changed their syllabus to use j's instead of i's now? When I did A-level I took my exams with OCR, and i'm pretty sure they did that.

Are you doing Maths at uni by any chance?

The dinner was a veggie one, complete with Quorn fillets!

Yep, on a Maths course before doing my PGCE. I love Maths, but I am more into working with less able children ... I would be very happy if they left school knowing how to add up their shopping bill, even if it's only an estimate ... can't see them ever being interested in Complex Numbers! x

lellie

jamiex wrote: »

Haven't some exam boards changed their syllabus to use j's instead of i's now? When I did A-level I took my exams with OCR, and i'm pretty sure they did that.

Are you doing Maths at uni by any chance?

I don't know about the exam boards, when I sat A-levels it was definitely i.

And yep, I am doing maths at uni, at least until my very last exam on the 30th of May! :eek:

EDIT: Dammit, I'm tr3mor, this is my girlfriends account!