Final answer:
There isn't a traditional fundamental frequency for the given function because the angular frequencies of the individual terms do not share a common divisor. The Fourier coefficients are simply the amplitudes and frequencies of the individual sine and cosine terms in the function.
Step-by-step explanation:
The given function x(t) = sin(2/3 t) + 2 cos(5/16 t) is a sum of two harmonic functions, each with its own frequency. The fundamental frequency ω₀ is the greatest common divisor of the angular frequencies of the individual terms, but since these frequencies (2/3 and 5/16) have no common divisor, we cannot speak of a fundamental frequency in the traditional sense of a Fourier series that assumes periodicity.
Nevertheless, we can present their Fourier coefficients as they stand because each term represents a single harmonic component.
For the trigonometric Fourier series, the coefficients for a single term A sin(ωt + φ) or A cos(ωt) are just the amplitude A and the angular frequency ω with which the sine or cosine function oscillates. In the case of x(t), we have an amplitude of 1 for the sine term with an angular frequency of 2/3, and an amplitude of 2 for the cosine term with an angular frequency of 5/16, with no phase shift in either term as they are already in standard form.