# New Plane And Spherical Trigonometry

New Plane And Spherical Trigonometry
Webster Wells
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7. Given A = 78^ 42' 53", b = .0520281.
8. Given a = .932368, c = 4.786723.
9. Given 5 = 72^8' 27", c = 27.28338.
10. Given J5 = 49* 25' 34", &=. 3376494.
Vn. QENERAL PROPERTIES OF TRIANQLEa ]j07. In any triangle, the mdes are proportional to the wms of theif egpoaUe angles.
L Topiove a:6 = 8injl:8m& (47) There will be two cases, according as the angles A and B are both acute (Fig. 1), or one of them obtuse (Fig. 2).
In each case, draw CD perpendicular to
... AB.
Then in each figure, CD = & sin^l (§ 4).
Also in Fig. 1, CD=^a sin B.
And in Fig. 2, CD=ra sin CBD = a sin (180** — S) = a sin^ (\$ 38).
& sin jii = a sin B.
Then in either case.
Whence by the theory of proportion, a: &=:sinj!i: sinJ3.
In like manner, 5 : o = sin B : sin C, and o: as sin C7: sin A (48) (49) 106L In any triangle, the gum of any two aides is to their difference as thetangent of half the sum of the opposite angles is to the tangent of half their .difference.
By (47), a:& = sinj!i:sinA Whence by composition and division, a + &:a — & = sin jii + sin^: sin jl — Bin& a + b sin jii + sin JB Or.

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