In planar trigonometry, there is a cosine formula
a2 = b2 + c2 – 2bc cos A. (B.1)
In spherical trigonometry, a similar formula exists:
cos a = cos b cos c + sin b sin c cos A. (B.2)
When the arcs are short and the spherical triangle approaches a planar triangle, Eq. B.2 reduces to Eq. B.1. In fact, for small arcs,
cos b & 1 – 1 b2, (B.3)
sin b & b, (B.4)
and so on. Substituting Eqs B.3 and B.4 into Eq. B.2 reduces it to Eq. B.1.
Here we give a simple proof of the cosine formula in spherical trigonometry by an analogy to that in planar trigonometry. To simplify notation, we set the radius of the sphere OA = OB = OC = 1. By extending line OB to intersect a line tangential to AB at a point D, we have
AD = tan c; OD = sec c.
Similarly, by extending line OC to intersect a line tengential to AC at at a point E, we have
AE = tan b; OE = sec b. (B.6)
From the planar triangle DAE, using the planar cosine formula,
DE2 = AD2 + AE2 – 2 AD • AE cos DAE = tan2 c + tan2 b — 2tan b tan c cos A.
Figure B.2 Derivation of cosine formula. The derivation is based on the projection of a spherical triangle onto a plane and the cosine formula in planar trigonometry. A straight line tangential to arc AB intersects the extension of line OB at D. Another straight line tangential to arc AC intersects the extension of line OC at E. By applying the cosine formula in planar trigonometry on triangles ODE and ADE, after some brief algebra, the corresponding cosine formula is obtained.
Similarly, for the planar triangle DOE,
DE2 = OD2 + OE2 — 2 OD • OE cos DDOE = sec2 c + sec2 b — 2 sec b sec c cos a.
Subtrating Eq. B.7 from B.8, and notice that
sec2 b = 1 + tan2 b, sec2 c = 1 + tan2 c,
2 — 2 sec b sec c cos a = —2tan b tan c cos A. Multiplying both sides with cos b cos c yields
cos a = cos b cos c + sin b sin c cos A. Similarly, for vertices B and C,
cos b = cos c cos a + sin c sin a cos B;
cos c = cos a cos b + sin a sin b cos C.