C3 Trig R sin/cos(theta + alpha) help?

Hi
Was wondering how you know which form to make it in?

ie.
looking at this question:

2cos3theta - 3 sin3theta = -1

would you put it in the form:

Rsin (3theta + alpha) and go from there or
Rcos(3theta + alpha)

I have a sheet with an example Rsin(theta + alpha) which I've just been following for another question, it works, but I don't understand why. I hate this topic. -.-

Reply 1

Student29

I think you use the latter equation...so:

2cos3theta - 3 sin3theta = Rcos(3theta + alpha)

Expand the second equation, equate the co-efficients, cancel out the identical parts, then figure what R and alpha is from there.

Reply 2

kej817

Original post by Student29

I know in this case you use Rcos, and what to do from there, but how did you know how to use Rcos in the first place instead of Rsin?

Reply 3

Student29

Because if you use the former equation you will not be able to cancel out the co-efficients as smoothly as you can with the latter equation?

As far as i am aware, generally:

if Asintheta + Bcostheta, then the equation Rsin (theta + alpha) must be applied.

If Acostheta + Bsintheta, then the equation Rcos(theta + alpha) must be applied.

as your equation starts with '2cos....' then the second rule is applied.

hope that explains it :redface:

Reply 4

kej817

but say you had the equation

2sintheta + 3costheta = 5

you could easily rearrange that to

3costheta + 2sintheta = 5

:s-smilie:

I dont understand why the order of them being different requires a completely different factor being taken out

Reply 5

Student29

Original post by kej817

but say you had the equation

2sintheta + 3costheta = 5

you could easily rearrange that to

3costheta + 2sintheta = 5

:s-smilie:

I dont understand why the order of them being different requires a completely different factor being taken out

Sorry i wrote the rules wrong...
its actually (according to my textbook):

Asintheta + Bcostheta, then the equation Rsin (theta + alpha) must be applied.

If Acostheta + Bsintheta, then the equation Rcos(theta - alpha) must be applied.

If Asintheta - Bcostheta, then the equation Rsin (theta - alpha) must be applied

If Acostheta - Bsintheta, then the equation Rcos(theta + alpha) must be applied.

Don't have time to explain it completely, but hopefully these rules can help you out!

Reply 6

kej817

Original post by Student29

Hey, thanks a lot, this will serve a lot of use for now anyway with my homework. :smile:

I'll try and get my teacher to explain it to me why it works some time soon.

Reply 7

mathflair

Original post by Student29

How is 'alpha' calculated for each of these equations?

Reply 8

james.h

Consider the addition formulae:

\sin(\theta \pm \alpha) = \sin(\theta)\cos(\alpha) \pm \cos(\theta)\sin(\alpha)

\cos(\theta \pm \alpha) = \cos(\theta)\cos(\alpha) \mp \sin(\theta)\sin(\alpha)

What you're doing in these sorts of questions is just working backwards. :yep:

Sorry, no time to elaborate at the moment; morning lecture.

Reply 9

Intriguing Alias

Original post by Student29

Those rules seem pretty arbitrary, for example you don't need to write +alpha or -alpha. Just always write +alpha and sometimes you'll get a negative value.

@OP Unless I'm missing something (if so, I only just got up so forgive me) but either one can work on any of the equations. After all every Cos graph is just a translated Sin graph etc.

The exam itself should tell you which form to write it in surely?

Reply 10

kurdishboy94

by using tangent rule of opposite over adjacent and we know our opposite and adjacent as we found out what the coefficients of our equation were then do arctan of opposite over adjacent and you get alpha.