# Determining Lat, Long using trigonometry

1. Mar 15, 2012

### Insanity

I am not sure if this is entirely possible, as my spherical trig skills are lacking.

Would it be possible to determine the Lat and Long coordinates for a position on the Earth knowing the following.

1. True North Pole is at 90°N
2. Geomagnetic North Pole is at 82.7°N, 114.4°W
3. Knowing the direction of True North, I can determine the angle separating the two at my position.

The vertices of the spherical triangle would be.
1. Point A would be the True North Pole.
2. Point B would be the Geomagnetic North Pole.
3. Point C would be my location.

The sides of the spherical triangle would be.
1. Line AB, or γ
2. Line BC, or α
3. Line AC, or β

Side γ has a known length, the distance from the True Pole to GM Pole, which is 820.76km. With Earth's radius at 6371km, this works out to be 0.13 radians, or 7.38°.

$sin C/sin γ = -1.07703 = sin A/sin α = sin B/sin β$ Correct?

Seems to me the other lengths should be able to be determined, and then perhaps the coordinates of Point C. Is that possible?

2. Mar 16, 2012

### tiny-tim

Hi Insanity!

(i'm not sure what your knowns are, but anyway …)

You can solve any spherical triangle by using the standard identities of spherical trigonometry …

3. Mar 16, 2012

### Insanity

Maybe I have the law of sines upside down?

C is the angle between vertices, γ is the length of the opposite side in degrees, so then

sin γ/sin C = -0.92848

I know the lat, long of two vertices, A and B, of the triangle, and the length of one side γ, and the angle of its opposite vertice C.