how to sketch the field lines of $F(x,y)=(\sin y,-\sin x)$?

Question

how to sketch the field lines of $\vec{F}(x,y)=(\sin y,-\sin x)$?

If i use the usual method,then i have $\frac{dx}{dt}=\sin y$ and $\frac{dy}{dt}=-\sin x$. To draw the field line, i have to solve $$\frac{dy}{dx}=-\frac{\sin x}{\sin y}$$

But if i solve this ODE, i get $\cos y=-\cos x +C \Rightarrow y=\cos^{-1}(-\cos x +C)$

Now, here comes the question, how do I draw the field line without using computer?

Or,is there any other way to sketch the field lines other than the method i used above?

Answer 1

Which information other than the diagram below, are you after?