Question

Find an identity for cos5x in terms of cosx.

Answer 1

cos(x+x+x+x+x) :)

Answer 2

I don't think that's what they had in mind. lol

Answer 3

:) start with cos(x + 4x) and turn it into:

cos(x)sin(4x) - cos(4x)sin(x) and work it all down to from there, if i see it right; but that is a pain

Answer 4

I've tried that and several other attempts. Perhaps its just because I hate arithmetic. the text has16cosx^5 - 20cosx^3 +5cosx

Answer 5

cos(x)sin(4x) - cos(4x)sin(x)

cos(x)[sin(x)sin(3x) + cos(x)cos(3x)] - [cos(x)cos(3x)-sin(x)sin(3x)]sin(x)

cos(x) - sin(x)
sin(x)sin(3x) + cos(x)cos(3x) cos(x)cos(3x)-sin(x)sin(3x)

...........................................................
cos(x)
sin(x) +cos(x)
sin(x)sin(2x) + cos(x)cos(2x) cos(x)cos(2x) - sin(x)sin(2x)

- sin(x)
cos(x) -sin(x)
cos(x)cos(2x) - sin(x)sin(2x) sin(x)sin(2x) + cos(x)cos(2x)
...............................................................

yeah, its a pain :)

Answer 6

Can you use the triple and double angle formulas to break it up?

I started, but my brain is hurting. Split it into
cos(3x + 2x) = cos(3x)cos(2x) - sin(3x)sin(2x).

I worked the first term using cos(3x)=cos^3(x) - 3sin^2(x)cos(x) and cos(2x)=cos^2(x)-sin^2(x) and got

8cos^5(x)-7cos^3(x)+3cos(x).

Haven't worked the second term though...but it looks promising, no?

Answer 7

It does. I haven't come up with coefficients that look that hopeful in many attempts

Answer 8

\[Cos[5 x] = 5 Cos[x] - 20 Cos[x]^3 + 16 Cos[x]^5 \]

Answer 9

Ok...my algebra is, I'm sure off, but for me it reduced to -8cos^5x - 12cos^3x +cosx which is not what your book says.

But I think the idea is sound...but my algebra is wonky.

Answer 10

Got it!

Answer 11

whew!

Answer 12

beautiful. thanks to both of you.

Answer 13

no problem. that was an outstanding refresher for me. it took putting down the pen and typing it out to keep my terms in place. :)

Answer 14

A Mathematica solution with comments is attached.

Answer 15

could you try again. All I got was a black box off screen. Thanks.

Answer 16

This site does't allow tiff files to be attached. I'll have to covert a tiff screen capture to jpg.

Answer 17

I'd appreciate it. I'll give you a medal in advance.

Answer 18

Thanks for the medal. A jpg is attached. Sorry for the delay. Had to bring up a virtualized version of Ubuntu Linux and then use the Terminal command, Convert.

Answer 19

There was an error in the last comment. Equate Cos[x]^5 to .... should have been typed,
Equate Cos[5x] to ... Correct pdf version is attached.