Derivative of sin(x) + cos(y) = 0.5 to find where gradient =0

Question

Take the graph $\sin(x) + \cos(y) =0.5$ (hint its really nasty) I want to find the point where the tangent would be parrallel to the x-axis which I assume means where the gradient = 0

However I have no idea how to do the derivative of $\sin(x) + \cos(y) =0.5$ Any help would be great. Thanks

Answer 1

Let $f(x,y)=\sin x+\cos y$. The tangent to a point of the curve $f(x,y)=\frac12$ will be horizontal when and only when the gradient of $f$ is vertivel, that is, when $\frac{\partial f}{\partial x}=0$. But $\frac{\partial f}{\partial x}(x,y)=\cos x$.

Answer 2

You take the derivative as follows:

Write $$\sin(x)+\cos(y)=\frac12\to \cos(x)-\sin(y)\,\frac{dy}{dx}=0$$

and draw $\dfrac{dy}{dx}$.