. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. PLANE TRIGONOMETRY. 55 But, tan. ( —-— ): tan. [ —-— J =+ : sin«4—(eq.(l), trig.) Comparing the two latter proportions (th. 6, b. 2), we have, iA+B A—B CB+AQ : CB—AC= tan. ( =-£=- ) : tan. ( —— \ Q. E. D PROPOSITION 8. Given the three sides of any plane triangle, to find some relationwhich they must bear to the sines and cosines of the respective angles. Let ABC be thetriangle, and let theperpendicular falleither upon, orwithout the base,as shown in
. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. PLANE TRIGONOMETRY. 55 But, tan. ( —-— ): tan. [ —-— J =+ : sin«4—(eq.(l), trig.) Comparing the two latter proportions (th. 6, b. 2), we have, iA+B A—B CB+AQ : CB—AC= tan. ( =-£=- ) : tan. ( —— \ Q. E. D PROPOSITION 8. Given the three sides of any plane triangle, to find some relationwhich they must bear to the sines and cosines of the respective angles. Let ABC be thetriangle, and let theperpendicular falleither upon, orwithout the base,as shown in thefigures; and byrecurring to theorem 38, book 1, we shall find. CD-. aa+5s 2a (1) Now, by proposition 3, trigonometry, we have,R : cos. (7=5 : CD ™, ,. yvT-v & cos. C .... Therefore, . CD= R (2) Equating these two values of CD, and reducing, we have,n R{a?-\-V—c>) . , In this expression we observe that the part of the numeratorwhich has the minus sign, is the side opposite to the angle; andthat the denominator is twice the rectangle of the sides adjacentto the angle. From these observations we at once draw the fol-lowing expressions for the cosine A, and cosine B. R^+c—a2)~2bc~ Thus, ~- R(a2+c2—P) 2ac (»)(p) 56 SURVEYING. As these expressions are not convenient for logarithmic compu-tation, we modify them as follows : If we put 2a=A, in equation (31), we have,cos.^4-{-l=2 ^4 In the preceding expression (n), if we consider radius, unity,and add 1 to both members, we shall have, cos. ^4 4 1=1Therefore, 2 \A— 2bc2bc-\-b2-\-c*—a? 2bc_(5+c)2_a2 2bcConsidering (b-\-c ) as one quantity, and observing that w
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