Plane And Spherical Trigonometry

Cover Plane And Spherical Trigonometry
Plane And Spherical Trigonometry
Leonard M Leonard Magruder Passano
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These formu- lae are simple and convenient, but it is unnecessary to bur- den the memory with them.
80. Cases 3 and 4, two sides and the included angle, two angles and the included side, are solved by means of Napier's Analogies, which we shall proceed to derive.
From (48), Art. 78, we may write A ^ B * tan ^- tan ^ = 2 2 sin(s-a) sin(s-6)' 112 SPHERICAL TRIGONOMETRY [X, 80 or, since, 2 _ sin (s - a) sin (s b) sin (s c) sins sin sin 2 2 _ sin (s - c) "2"' ~^~ sins cos cos 2 2 Whence, sin - sin
..._ 1 _ _2 _ 2 _ 1 _ sin (s - c) cos^cos*' sins 2 2 or, cos cos -- sin sin 22 22 _ sins-sin(s-c) eosfcosf = That is, cos i (A + B) _ 2 cos |(2 s c) sin | c . 4 J5 sins COS C S ~ Whence, since 2s c = a-|-6-|-c c = a + 6, cos J (^ + B) _ 2 cos i (a 4- 6) sin i c, . ~ ^|^f = S5T ^' Also, from (a) above, sin sin -, , _2 _ 2 _ sm(s-c) += which being transformed in the same manner gives cos i (A B) __ 2 sin|(q + 6) cos j-c "~Z 5~ : sins cos cos 2 2 X, 80] OBLIQUE SPHERICAL TRIANGLES 113 Dividing (ft) by (y) we have cosiQi + B) = tan|c (49) cos|(4-5) tan(a+&) Again, from (48) Art.

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