Particle disintegration problem from Landafshitz

[Geofleur]

I've been stuck on problem 3 from Ch. 4 of Landau and Lifgarbagez's Mechanics for more than a week now. It's not a homework problem - I'm working through Landau and Lifgarbagez on my own.

A particle at rest disintegrates into two, and the problem is to find the range of values that the angle θ between the resulting particles can take, where θ is measured from the lab system.

There are formulas that give tan(θ1) and tan(θ2) in terms of the angle θ0 of particle 1 with respect to the x-axis in the center of mass frame, and θ1,2 are the angles that particles 1,2 make with the x-axis in the lab frame.

The separation angle is thus θ = θ1+θ2 and I calculate the tangent of this angle by using the formula tan(θ1+θ2) = (tan θ1 + tan θ2)/(1 - tan θ1 tan θ2). Taking the derivative of this expression, setting the result to zero, and solving for θ0 gives me an answer in terms of cos θ0, but the answer in the book is in terms of sin θ0; trying to use cos θ0 = √(1 - sin θ0) does not help at all!

Is there anyone here that has worked this problem and knows where their answer comes from? Or maybe a different tack to try?

 

[Geofleur]

I found the problem. My diagram was wrong, so that I was using pi minus the angle I really needed in one place! How embarrassing...