Placing four points on a sphere with maximum spacing

This thought just popped into my head and i gave the problem a shot and i was wondering if anybody could let me know a) if i got the correct answer and b) how it SHOULD be solved (I haven't really done any work with three dimensional math).

So the general idea is that you have a sphere (radius of 1) and on it you have to place 4 points allowing for the maximum amount of space in between each point.

My first thought was that the basic idea would be to have the points evenly spaced (0, 90, 180, 270 degrees) along a cross section then move one pair of opposites up along a z axis and the other two down by the same angle. If this is wrong then you can skip the rest and just let me know what i should have done.

Otherwise....

My idea to find the angle was to set up the equation that the distance between two points moved the same direction on the z axis should equal the distance between one point moved up and another moved down.

Now, let me note here, I'm not exactly confident in using measurements along the outside of the sphere so all my equations were for the distance through the sphere, if my logic is right this shouldn't change anything, again, if this is wrong I'd appreciate an explanation why.

anyways, for the distance between two moved the same direction on the z axis i got 2cosx (with x being the angle every point moved along z axis)

for the distance between a point moved up and one moved down i got sqrt(2((sinx)^2+1)).

when i set these equal i got x to be about 35.26 degrees and the distance through the sphere to be 1.63

Any knowledge about a problem like this would be greatly appreciated.