Detect when a point belongs to a bounding box with distances I have a box with known bounding coordinates (latitudes and longitudes): latN, latS, lonW, lonE.
I have a mystery point P with unknown coordinates. The only data available is the distance from P to any point p. dist(p,P).`
I need a function that tells me whether this point is inside or outside the box.

Measure distances from the four corners of the rectangle.
Consider the four triangles formed from P and the four sides of the rectangle.
Apply the Law of Cosines to measure the angles of the four triangles next to the sides of the rectangle.
If any is larger than 90 degrees then you are outside. Otherwise you are inside.
EDIT 1:
Suppose your rectangle is $EFGH$ where the sides are $d(EF)=d(HG)=x$ and $d(FG)=d(EH)=y$.
Let the distance of $P$ from $E,F,G,H$ be $e,f,g,h$ respectively. 
Now by Law of Cosines applied to  angle $EHP=\alpha$ we have $\cos (\alpha) = (y^2+h^2-e^2)/(2yh)$. If the angle is larger than $90^\circ$ then $y^2+h^2-e^2<0$. We need to apply a similar check 8 times. If any of the expressions is negative then we are outside. Else we are inside (or on boundary).  So you have to check the sign of  8 expressions.
If either of the following 
$y^2+h^2-e^2$,  $y^2+e^2-h^2$, $x^2+e^2-f^2$, $x^2+f^2-e^2$, $y^2+f^2-g^2$, $y^2+g^2-f^2$, $x^2+g^2-h^2$, $x^2+h^2-g^2$
is negative $P$ is outside.
EDIT 2:
In case efficiency is a consideration: To test that $P$ is inside, or on the border, checking with respect to 3 sides will do. So it takes 6 inequalities. For example
$y^2+h^2-e^2\ge 0$, and  $y^2+e^2-h^2\ge 0$, and 
$x^2+e^2-f^2\ge 0$, and $x^2+f^2-e^2\ge 0$, and 
$y^2+f^2-g^2\ge 0$, and $y^2+g^2-f^2\ge 0$.
EDIT 3:
A similar approach will work for checking with respect to a convex polygon.
