Answer
$t_1=\frac{1}{8}arctan~\frac{1}{3}\approx0.040~s$
$t_2=\frac{1}{8}arctan~\frac{1}{3}+\frac{1}{8}\pi\approx0.433~s$
$t_3=\frac{1}{8}arctan~\frac{1}{3}+\frac{1}{4}\pi\approx0.826~s$
Work Step by Step
$y=\frac{1}{12}(cos~8t-3~sin~8t)$
$0=\frac{1}{12}(cos~8t-3~sin~8t)$
$0=cos~8t-3~sin~8t$
$3~sin~8t=cos~8t$
$\frac{sin~8t}{cos~8t}=\frac{1}{3}$
$tan~8t=\frac{1}{3}$
$tan$ has a period of $\pi$:
$8t=arctan~\frac{1}{3}+n\pi$, where $n$ is an integer.
$t=\frac{1}{8}arctan~\frac{1}{3}+\frac{n}{8}\pi$
For $n=0$:
$t=\frac{1}{8}arctan~\frac{1}{3}\approx0.040$
For $n=1$:
$t=\frac{1}{8}arctan~\frac{1}{3}+\frac{1}{8}\pi\approx0.433$
For $n=2$:
$t=\frac{1}{8}arctan~\frac{1}{3}+\frac{2}{8}\pi\approx0.826$
