P$-3\mathrm{.}2$ A signal $x(t)$ has the two$-$sided spectrum representation shown in Fig. P$-3\mathrm{.}2.$

$($a$)$ Write an equation for $x(t)$ as a sum of cosines.

$($b$)$ Is $x(t)$ a periodic signal? If so$,$ determine its fundamental period and its fundamental frequency.

$($c$)$ Explain why "negative" frequencies are needed in the spectrum.