Answer
$$
x=\sin t + \sin1.5 t, \quad y=\cos t, \quad 0 \leq t \leq 4 \pi
$$the length of the curve is
$$
\begin{aligned}
L &=\int_{0}^{4\pi} \sqrt{(d x / d t)^{2}+(d y / d t)^{2}} d t \\
& \approx 16.7102
\end{aligned}
$$
Work Step by Step
$$
x=\sin t + \sin1.5 t, \quad y=\cos t, \quad 0 \leq t \leq 4 \pi
$$
then
$$
d x / d t=\cos t + 1.5 \cos 1.5 t , \quad d y / d t=- \sin t
$$
so
$$
(d x / d t)^{2}+(d y / d t)^{2}=\cos ^{2} t+3 \cos t \cos 1.5 t+ \\
\quad \quad \quad \quad \quad \quad \quad \quad \quad + 2.25 \cos ^{2} 1.5 t+\sin ^{2} t
$$
Thus the length of the curve is
$$
\begin{aligned}
L &=\int_{0}^{4\pi} \sqrt{(d x / d t)^{2}+(d y / d t)^{2}} d t \\
&=\int_{0}^{4 \pi} \sqrt{1+3 \cos t \cos 1.5 t+2.25 \cos ^{2} 1.5 t} d t \\
& \approx 16.7102
\end{aligned}
$$
