How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy \[1-3\sin\theta+5\cos3\theta = 0?\]
$\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8$
We rearrange to get \[5\cos3\theta = 3\sin\theta-1.\]
We can graph two functions in this case: $y=5\cos{3x}$ and $y=3\sin{x} -1$.
Using transformation of functions, we know that $5\cos{3x}$ is just a cosine function with amplitude $5$ and period $\frac{2\pi}{3}$. Similarly, $3\sin{x} -1$ is just a sine function with amplitude $3$ and shifted $1$ unit downward:

So, we have $\boxed{\textbf{(D) }6}$ solutions. 
~Jamess2022 (burntTacos)