Plus sign in Einstein's 1905 Doppler formula for opposite light rays

[SiennaTheGr8]

Link: https://www.fourmilab.ch/etexts/einstein/E_mc2/www/

The only part I'm having trouble with is how he gets the plus sign in that 1+(v/c)cos(φ) numerator for the "other" light ray (emitted in the opposite direction of the first).

My understanding is that the φ he uses in his general Doppler equation represents the angle formed by the source/receiver line-of-sight (as seen by the source at the time of emission) and the receiver's direction of motion (as measured in the source frame). If that's right, then the plus sign in that numerator is equivalent to taking the cosine of (φ + 180°) rather than φ for the "other" light ray.

So I'm 99% sure that it's an implied symmetry argument, but I don't know how to demonstrate its validity mathematically.

Can anyone help me out?

 

[SiennaTheGr8]

I think maybe I've misunderstood what the angle represents, and that this is much simpler than I suspected.

 

[stevendaryl]

Link: https://www.fourmilab.ch/etexts/einstein/E_mc2/www/

The only part I'm having trouble with is how he gets the plus sign in that 1+(v/c)cos(φ) numerator for the "other" light ray (emitted in the opposite direction of the first).

My understanding is that the φ he uses in his general Doppler equation represents the angle formed by the source/receiver line-of-sight (as seen by the source at the time of emission) and the receiver's direction of motion (as measured in the source frame). If that's right, then the plus sign in that numerator is equivalent to taking the cosine of (φ + 180°) rather than φ for the "other" light ray.

So I'm 99% sure that it's an implied symmetry argument, but I don't know how to demonstrate its validity mathematically.

Can anyone help me out?

If two photons are sent in opposite directions, and one makes an angle of $\varphi$ with respect to the x-axis, then the other makes an angle of $\varphi + 180^o$ with respect to the x axis. So for the transformation of frequency, etc., for the second photon, you just replace $\varphi$ by $\varphi + 180^o$.