Astronomy Principles and Practice Fourth Edition.pdf

Spherical trigonometry 53
Figure 7.6. Proof of the cosine formula.
Draw tangents at A to the great circle arcs AB and AC to meet the radii OB and OC produced
at D and E respectively. Join D and E. JoinO and A, O being the centre of the sphere.
Then triangles ADE, ODE are plane triangles, though not in the same plane. By definition,
DAE is the spherical angle A. Also DOE is the angle subtended by the arc BC. Hence,
DOE = a.
In △DAE we have
DE 2 = AD 2 + AE 2 − 2AD.AE cos A. (7.10)
In △DOE we have
DE 2 = OD 2 + OE 2 − 2OD.OEcos a. (7.11)
Hence, by subtraction
2OD.OEcos a = (OD 2 − AD 2 ) + (OE 2 − AE 2 ) + 2AD.AE cos A. (7.12)
Now △DAO is a plane triangle right-angled at A, sinceDA is a tangent to the circle AB at A
and OAis a radius of that circle. Hence, by Pythagoras,
OD 2 − AD 2 = AO 2 .
Similarly, by considering △OAE, it is seen that
Hence, equation (7.12) becomes
OE 2 − AE 2 = AO 2 .
OD.OEcos a = AO 2 + AD.AE cos A
or
cos a = OA
OD . OA
OE + AD
OD . AE cos A.
OE
In △DAO, OA/OD is the cosine of the angle DOA while AD/OD is the sine of that angle.
But DOA = c. Hence, OA/OD = cos c; AD/OD = sin c.
Similarly OA/OE = cos b; AE/OE = sin b.
Hence,
cos a = cos b cos c + sin b sin c cos A. (7.4)