How to get the angle of a mach cone given two other angles? 
I have the following problem at hand: I know the angle Phi of a cone, which is the angle between the outer boundary and the X axis. The outer boundary of the cone intersects the wavefronts(black circles) of a mach cone at the yelloq point. As for every point inside the mach cone there a two wavefronts arriving at the point I am only interested in the intersection with the wavefront with the smaller circle radius (or simply speaking only intersections with the left half of the circle counts).
Then I also have an angle Alpha, which describes the angle between the Y Axis and another line which touches the wavefronts at the yellow point tangentially.
Given these two angles how do I calculate the angle θ of the mach cone?

Analyze the following figure:

Let $T$ the wave period, $c$ the wave speed and $v$ the font speed.
From Physics we know that:
$$c=v \sin \theta. \tag{1}$$
From triangles AOY and QYO we have:
$$\lambda=\alpha-\varphi. \tag{2}$$
Applying the law of sines in triangle AOY we obtain:
$$\frac{c}{v}\sin\lambda=\sin\varphi. \tag{3}$$
Substituting $(1)$ and $(2)$ in $(3)$ we get:
$$\sin\theta=\frac{\sin\varphi}{\sin(\alpha-\varphi)}.\tag{4}$$
