Answer
In December, the average monthly temperature in Phoenix is $55^\circ F$.
Work Step by Step
$$f(x)=19.5\cos[\frac{\pi}{6}(x-7)]+70.5$$
The question asks when the average monthly temperature in Phoenix is $55^\circ F$
In other words, find $x$ so that $f(x)=55$.
Therefore, we can replace $f(x)=55$ to the given equation and find $x$:
$$19.5\cos[\frac{\pi}{6}(x-7)]+70.5=55$$
$$19.5\cos[\frac{\pi}{6}(x-7)]=-15.5$$
$$\cos[\frac{\pi}{6}(x-7)]=-\frac{15.5}{19.5}=-\frac{31}{39}$$
$$\frac{\pi}{6}(x-7)=\cos^{-1}(-\frac{31}{39})\approx2.4896$$
$$x-7=2.4896\times\frac{6}{\pi}$$
$$x-7\approx4.7548$$
$$x=11.7548\approx12$$
$x=12$ refers to the month December. Thus, in December, the average monthly temperature in Phoenix is $55^\circ F$.
