Law of Cosines for Spherical Triangles?

Question

Can you help me with the Law of Cosines for spherical triangles please?
(A) Draw a spherical triangle and label each vertex by A, B, and C. Then connect each vertex by a radius to the center O of the sphere. Now draw tangent lines to the sides a and b of the triangle that go through C. Extend the lines OA and OB to intersect the tangent lines at P and Q, respectively. List the plane right triangles. Determine the measures of the central angles.
(B) Apply the Law of Cosines to triangles OPQ and CPQ to find 2 expressions for the length of PQ.
(C) Subtract the expressions in part (B) from each other. Solve for the term containing cos C.
(D) Use the Pythagorean theorem to find another value for OQ^2-CQ^2 and OP^2-CP^2. Now solve for cos C.
(E) Replacing the ratios in part (D) by the cosines of the sides of the spherical triangle, you should now have the Law of Cosines for spherical triangles: cos C=cos Acos B+sin Asin Bcos C.
I got through parts A B C D, but I need help at E

I got to
Cos C=((OC^2)/(OP)(OQ))-Cos O
Can you help me from there?

Answer 1

Hi There, I can't help you, that question looks like it would take ages, I suggest you look at more examples and work off them. I did a search for you and found some good links.


If you've done it all except for part E then it seems like it should be simple substitution and then some algebra using trig ratios.