Final answer:
The fundamental frequency ω0 is the lowest frequency of oscillation and the Fourier series coefficients are constants that can be directly observed from the function given, which are 1, 5, and 9 for the cosine and sine terms respectively.
Step-by-step explanation:
The fundamental frequency ω0 for a wave is the lowest frequency at which the wave oscillates. In the case of trigonometric Fourier series of x(t), this frequency corresponds to the lowest common multiple of the angular frequencies present in the function. The given function x(t) = cos(1/3 t) + 5 sin(1/3 t) + 9 cos(1/7 t) consists of two angular frequencies: 1/3 and 1/7. To find ω0, identify the periods of these components as Τ1 = 2π/(1/3) and Τ2 = 2π/(1/7) and find their least common multiple.
The coefficients of the Fourier series are directly observable from the given function: the coefficient of the cosine term with angular frequency 1/3 is 1, the coefficient of the sine term with angular frequency 1/3 is 5, and the coefficient of the cosine term with angular frequency 1/7 is 9. There are no coefficients other than these constants since the given function only consists of these harmonic terms.