Relating flux and angular velocity of constantly rotating coil with induced voltage

The flux and angular position of the graph has been recorded as shown in the graph.
and so is induced voltage and angular position.

and my teacher derived the relation between induced voltage and angular position as such:

My question is how he came to this relation. All he said was flux and angular position has a linear relationship we can write the equation for induced voltage as such.

If so does it imply that Flux= BAphi such that derivative of it will be (Flux)’=BA(phi)’ and how can I say Flux =BAphi from the flux and angle graph?
I have tried to find linear equations for the different slope of the graph and couldn’t arrive at the derivation he made.

My approach:

I find the function for every piecewise continuous graph which gives me the following result: Φ=-B∙A, Φ=((2B∙A)/(π-2θ))∙(φ-π/2), Φ=B∙A,
Φ=-((2B∙A)/(π-2θ))∙(φ-3π/2), Φ=-B∙A and let me find the derivative of flux with respect to angle first which gives me:

dΦ/dϕ=(2B∙A)/(π-2θ), dΦ/dϕ=-(2B∙A)/(π-2θ) for two slopes
and derivative of flux w.r.t time comes out to be:
(dΦ/dϕ)∙(dϕ/dt)=((2B∙A)/(π-2θ))∙dϕ/dt, (dΦ/dϕ)∙(dϕ/dt)=-((2B∙A)/(π-2θ))∙dϕ/dt, where dϕ/dt=2πn.
and it still isn’t equal to what the professor derived.
I need insight on it.
Thank you.