How can we find the number of roots of $a\sin \theta+b\cos \theta-\sin\theta\cos\theta=0$ in $[0,2\pi)$.
Taking $t=\tan \frac{\theta}{2}$, we may change the above equation into a fourth order system $f(t)=bt^4-2(a-1)t^3-2(a+1)t-b$. And find the number of roots of $f$.