3 The surface \(S\) has equation \(z = \frac { x } { y } \sin y + \frac { y } { x } \cos x\) where \(0 < x \leqslant \pi\) and \(0 < y \leqslant \pi\).
- Find
- \(\frac { \partial z } { \partial x }\),
- \(\frac { \partial z } { \partial y }\).
- Determine the equation of the tangent plane to \(S\) at the point \(A\) where \(x = y = \frac { 1 } { 4 } \pi\). Give your answer in the form \(a x + b y + c z = d\) where \(a , b , c\) and \(d\) are exact constants.
- Write down a normal vector to \(S\) at \(A\).