Simple (or should be) Trigonomic question

[Yay]

Homework Statement

I'm making a program that tracks an object moving along the surface of a sphere (earth). I'm using a set of Cartesian coordinates who origin is the center of the Earth and Z directed through the north pole, X the greenwich meridian at the equator, and Y is 90 degrees to everyone else.

Given a Lat/Long value I can convert into a position vector by doing:

phi = (90 - lat)*pi/180
theta = long*pi/180

vector.x = sin(phi)*cos(theta)
vector.y = sin(phi)*sin(theta)
vector.z = cos(phi)

After moving my object around I need to convert it back into lat / long. At the moment I'm just trying to take a position in Lat/long, convert it into my position vector, draw it on a sphere , then convert it back.

I know my conversion TO the position vector is working properly since it points to the right spot on a sphere. What I can't get right is the conversion FROM the vector back to lat long.

Homework Equations

See above.

The Attempt at a Solution

I thought I could find phi from the Z component, then solve X and Y simultaneously for theta. So since Z = cos(phi), I thought easy phi = 1/cos(Z).

Only that didn't work. Sat down, thought for a bit, played around with cos and sec, and realized cos and sec aren't inverse operations (which I thought for sure they were) so sec(cos(a)) != a.

There's a simple little rule for finding going from A = cos(angle), to an expression for the angle, when you know A. Can someone please tell me what it is ? I've spent hours on something I thought I learned back in Highschool !

 

[berkeman]

Is there a reason you aren't using spherical coordinates instead of cartesian coordinates? Seems like you're making it extra hard on yourself.

Here are some trig notes from wikipedia if they help:

http://en.wikipedia.org/wiki/Trig

.

 

[Yay]

We were told to use Cartesian, not spherical. Gonna have another read through the wiki page, its got to be there somewhere.

 

[Yay]

I worked it out. Turns out sec is the reciprocal of cos, arccos is the inverse function. I'm annoyed I forgot something so basic, but at least its all working now.