The Sinusoidal Projection is a coordinate transform often used to map geographical coordinates on to a flat surface in order to generate a map. It is typically stated that this projection is area-preserving, meaning that while shapes and angles may be distorted, the ratio of the area of any region calculated in the projected coordinates to the area measured on the surface of the globe will be constant with respect to position on the globe.
The equations relating projected coordinates to geographical coordinates are:
$\begin{eqnarray} x &= &\lambda \cos{\phi} \\ y &= &\phi \end{eqnarray} $
where $\lambda$ is longitude and $\phi$ is latitude.
I wanted to check this myself. I recall (correctly, I hope) that the ratio by which a coordinate transform changes area is given by the magnitude of the determinant of the jacobian matrix of that transformation. For the transform I defined above, this is pretty trivially $\cos{\phi}$, which would imply that the area of a region in projected coordinates would vary depending on it's latitude.
So, I am confused. I clearly misunderstand something or have calculated something incorrectly, because my results would indicate that the sinusoidal projection is trivially not area preserving. What is my error, and how would I have correctly done this calculation?