Final answer:
The message signal m(t) for the FM signal is 2sin(wₘt + θ), and the frequency deviation Δf is 2 Hz.
Step-by-step explanation:
To determine the message signal m(t) and the frequency deviation Δf for the angle modulated signal given by v(t) = 4cos[wt + 2sin(wₘt + θ)], assuming it represents an FM (Frequency Modulated) signal, we first look at the inner argument of the cosine function, which pertains to the instantaneous phase of the signal. In FM, the instantaneous frequency is the derivative of the instantaneous phase with respect to time. Therefore, any function inside the cosine that is not a linear function of time t represents the modulation induced by the message signal m(t).
In this case, the message signal m(t) can be extracted as the function that is being multiplied by time in the argument of the sine function inside the cosine. This yields m(t) = 2sin(wₘt + θ). The peak deviation of the frequency from the center frequency (frequency deviation Δf) can be found by taking the maximum rate of change of the phase with respect to time, which would be the amplitude of the message signal m(t), thus Δf = 2 Hz. This frequency deviation is crucial for demodulating the FM signal and retrieving the original message.