Multiplying Cosines: Learn How to Represent x(t)

[SMOF]

Hello,

I hope everyone is well.

Also, I hope someone can help me understand something. I am trying to understand how cos(ω₁t) times cos(ω₂t) can be represented as

x(t) 1/2 (cos(ω₂ - ω₁)t) + 1/2 (cos(ω₂ + ω₁)t)

Thanks in advance for any help.

Seán

 

[Mentallic]

Try expanding cos(A+B) and cos(A-B)

 

[SMOF]

Hello,

Thanks for the reply.

So, cos(A+B) = cosA cosB - sinA sinB, and cos(A-B) = cosA cosB + sinA sinB.

But I don't see how this gets me to the solution I need, since it doesn't have a sin component.

But maybe I am just being stupid here.

Seán

 

[SMOF]

Oh, I may have got it with the Product to Sum Identity.

Seán

 

[Mentallic]

Sorry about the late reply, I am on holidays.
Ok notice that the cos(A+B) has a sinAsinB component and cos(A-B) has a -sinAsinB component, so why not add these two cosines sums to get rid of the sinAsinB?

 

[SMOF]

Hello.

That will just leave me with 2(cosAcosB) ...yea?

Seán

 

[HallsofIvy]

Hello,

Thanks for the reply.

So, cos(A+B) = cosA cosB - sinA sinB, and cos(A-B) = cosA cosB + sinA sinB.

But I don't see how this gets me to the solution I need, since it doesn't have a sin component.

But maybe I am just being stupid here.

Seán

So add them!

 

[Mentallic]

Hello.

That will just leave me with 2(cosAcosB) ...yea?

Seán

Right, and you want cosAcosB, so...

 

[SMOF]

Hello.

Yea, so, I divide by 2 ...or, multiply by a half ...I think I have it from here.

Many thanks for your help :)

Seán

 

[Mentallic]

Hello.

Yea, so, I divide by 2 ...or, multiply by a half ...I think I have it from here.

Many thanks for your help :)

Seán

Yep that's it

You're welcome, and good luck with your physics!