Cosine Formula

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)

2

and

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

Подпись: (B.5)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.

image825Similarly, for the planar triangle DOE,

Подпись: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,

we obtain

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.

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