Finding the maximum phase and the maximum frequency deviation of an angle-modulated signal

I have an angle-modulated signal: $$s(t)=10\left[\cos\left(10^8\pi · t + 5\sin\left(2\pi\cdot10^3·t\right)\right)\right]$$

How would I go about finding the maximum phase and the maximum frequency deviation?

• well, you look at the phase of the signal and find the maximum deviation of that? Feb 8 '21 at 16:53
• @MarcusMüller Ok, I guess I should clarify as I'm having trouble finding the phase, as this is different from the general formula of a sin or cos function, how would I go about finding the phase? Feb 8 '21 at 18:05
• huh? this doesn't seem any different? Please edit your question to explain explicitly what you consider the "general formula", and how this formula doesn't fit that. Feb 8 '21 at 18:28

I think this is treating the $$10^8\pi\cdot t$$ as the carrier frequency and the $$5 sin(2\pi\cdot 10^3 t)$$ as the modulation. So the maximum phase difference of this signal to the carrier without modulation would be $$\pm5$$ radians.
To compute the change in frequency you need to find the compute $$\frac{d}{dt}$$ of the argument of the cos (divided by $$2\pi$$) and subtract the carrier frequency.
$$\frac{d}{dt} 10^8\pi\cdot t + 5 sin(2\pi\cdot 10^3 t) = 10^8\pi + 2\cdot 5\cdot 10^3\pi cos(2\pi\cdot 10^3 t)$$
Divide by $$2\pi$$ to get Hz instead of radians.
$$Frequency = 5\cdot 10^7 + 5\cdot 10^3 cos(2\pi \cdot 10^3t)$$
Subtract off the carrier of 50MHz and you have a maximum frequency excursion of $$\pm 5$$ k Hz