Lest anyone feel that the problem is artificial, of course it is! But not unrealistic! The data shown could easily occur in control experiments; steady state controlled input/output responses and sinusoids are natural consequences of control situations. The problem exposes a basic failing in our control teaching. We need "real" data problems! The counterpart to the old fashioned word problems? Whose difficulty exposed limits to our teaching of algebra?
As with any good teaching problem, the statement data (distilled from earlier course problems) has been selected to avoid unnecessary computational complication: The sinusoids are drawn about separate centered dashed lines. Their common period estimate is about 6.2+ or 2 π, so that the frequency is 1/(2 π) and ω is 1. The phase shift (φ) estimates to 2.1-, i.e. 2/3 π or 120 Degrees. (From prior course experience, the students would have expected this.) The two sinusoids both have estimated amplitude of about: 1.0 (with different offset).
The data gives combined* response data, in addition to the sinusoids, for two steady state input situations [not just one. An integrator under two different initial conditions (perhaps controlled to two different set points) is one of the problem tricks; but different phases of a very low frequency sinusoid wouldn't have made a material difference in the answer.]. Thus (with the operable word: estimate): b=0 and c=.4. The sinusoidal input/output data allows k and a to be solved from the gain and phase difference shown. φ = 120 Degrees [tan(φ) = 1.73] gives a = 1.73.
In the original equation, then |k/((j ω+a)j ω)| equals 1, |j ω+a|2 = (1+1.732) = 4, so k/2 = 1 or k = 2.
Despite the trick, the problem can actually be solved blindly from appropriate equations.
* From linear superposition, which should be obvious in practice, even though it may not be intuitive on a first exposure to this "real" data in an academic context.
When I first encountered this student frequency response difficulty, I ended up reversing the (theory first, mechanics second) order of teaching the material. Further, I ended up black-boarding exercises where we plotted different frequency input/output sinusoids for simple (e.g. lag) process models, having the students spell out the meaning of amplitude and phase changes. How "grade school"; but it addressed a real student need. This led to simpler versions of the problem presented here, to convert the material to more formal and testable form. And of course one gets carried away, witness the current exercise!