Solving a system of two trigonometric equations

[pjeuck]

So, in our Control Engineering classes, in the presentation about synthesizing a regulator using Bode plot (using lead compensation and lag compensation), we have the following system of equations:

omega_lb1 / omega_ln1 = 12.6
atan(2 / omega_lb1) - atan(2 / omega_ln1) = -6 deg

Allegedly, the solution is:

omega_lb1 = 0.2 rad/sec
omega_ln1 = 0.016 rad/sec

But how can that system of equations be solved?

I worked it all out going back to fundamental trigonometric definitions.  It does work out to be a complex solution with a fourth order quadratic equation to solve.  I ran through the solution approach quickly and found it can be done.  I'll write it all out tomorrow and photograph it and then provide the link.  I will be going to a doctor so I might have it ready tomorrow but within a day to two I link the full solution.   You can't solve it like its a system of linear equations.  But the information given in the problem is enough.