Items related to Elements of Analytic Trigonometry, Plane and Spherical
Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1826 ...ways. These will be obvious, when we consider that the formulae Q, No. 4 and S, give two different values for cosine B, the one in the sine, the other in the cosine, of the half angle. Taking, then, in the first cos B = 1-2 sin8 4 B And substituting this value in the equation for cosine B, we 62 = c3 + o3--2ac + 4.c.a.sin2 f B = (a CO e) + 4.a.c.sin3 i B And using (a co c) as a factor common to both the right hand . /, 4.c.o.sina i B Extracting the x i 4-.C.O.MD? J B- & = (o CO c) I 1 + 1--I V (a CO c)a / It may be that, by the method used in the preceding Problem, this formula may be adapted to the use of logarithms, employing the that, according to C, No. 8, we 1 cos x Second Transformation. cos B = 2 cos " i B'--1 and substituting this value in No. 5, the formula 6- = a" + c- + 2ae--4.o.c.cos3 4 B And by processes exactly similar to those used for the formula 6, it will finally i-(. +.)(.-. ) 9 Here we have, by the same means as in the foregoing transformation, representing the second term under the radical as the square of a siwe or cosine, according to C, No. 4 or 5, and b y an analogous 4.e.-. cos3 4 B (o + e)- or 2 cos 4 B cos a; = (a.e)s 10 a + e And 6 = (c-f-o) sin a; 11 If the distances, a and c, were given in logarithms, (as may occur in astronomy, ) it will be ebserved, that in applying to the part (o c), in the two formulae 8 and 11, the same mode of transformation for the application of an auxiliary angle, by writing (a c) = a I 1 --1 in both the places where it occurs, these formulae will be transformed, and fitted for that use, upon the same principles, every thing else remaining as above...
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