Hi, I am trying to perform some triangulation on a sphere. I have 3 distinct points of which I know the lat/long positions. From my position I can measure the angles between between the 3 points as I see them. Using the cosine rule I came up with three equations to find the three unknown distances from the points (so I can triangulate).

Known values are d,e,f,alpha,beta. Can anyone help me unwrap these three equations to find the three unknowns a,b and c? Have tried and ended up with a mess on the page

cos f = cos a cos c + sin a sin c cos(alpha + beta)

cos d = cos a cos b + sin a sin b cos alpha

cos e = cos b cos c + sin b sin c cos beta