GR, small expansion, (perihelion derivation)

[binbagsss]

Homework Statement

Hi I am looking at the attached as part of the derivation and am stuck on how we go from 18 to 19

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Homework Equations

Above below

The Attempt at a Solution

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I'm pretty stuck. Lambda is small and not sin so can't see why one would expand out sine in a Taylor, though this seems to be one of the only possibilities I can see and then using a cosines trigomemtric identity, think this would also need expanding out both sin and cos as functions off phi And then regathering for the use of the cosine identity. However as I said small lambda not phi so don't really understand

Thanks

 

[Orodruin]

Hint: Use a trigonometric identity.

 

[binbagsss]

Hint: Use a trigonometric identity.

My attempt mentioned this. So it's not cos ( a + b ) ? As I've said above, we are expanding small lambda not phi, so don't really understand how this would apply. Ta .

 

[Orodruin]

So it's not cos ( a + b ) ? As I've said above, we are expanding small lambda not phi, so don't really understand how this would apply.

Yes it is. You only expand the functions where the phase is multiplied by $\lambda$.

 

[strangerep]

[...] small lambda not phi so don't really understand

Alternate hint #1: To see that (18) does indeed imply (19), expand the cos in (19) as a Taylor series in $\lambda$.

Alternate hint #2: If you didn't already know (19), do as Orodruin suggested, and then consider $~\cos(\lambda) \approx ~?~$ and $~\sin(\lambda) \approx ~?~$.