. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . = PB - PC - PD*. • • to n terms. Tor, put OB = a, OP = x, and 6 = -; then n PB2 = OP2 + OB2 - 2 OP • OB cos 9 = x2 + a2 — 2ax cos - (1) n v PC2 = OP2 + OC2 - 2 OP . OC cos a + 2ir n = x2 + a2-2ax cos a + 2?r: and so on . (2)n Multiplying (1), (2), (3),— together, we havePB2 • PC2 • to n terms = (x? - 2ax cos - + a2Yar* -2ax cos a + 2ir + ^ x(tf-2axcos^+^+a2\.- = a?n-2 anxn cos a + a2n . [by (4) of Art. 169] = O^ + OB2n ... (3) which proves the propositi
. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . = PB - PC - PD*. • • to n terms. Tor, put OB = a, OP = x, and 6 = -; then n PB2 = OP2 + OB2 - 2 OP • OB cos 9 = x2 + a2 — 2ax cos - (1) n v PC2 = OP2 + OC2 - 2 OP . OC cos a + 2ir n = x2 + a2-2ax cos a + 2?r: and so on . (2)n Multiplying (1), (2), (3),— together, we havePB2 • PC2 • to n terms = (x? - 2ax cos - + a2Yar* -2ax cos a + 2ir + ^ x(tf-2axcos^+^+a2\.- = a?n-2 anxn cos a + a2n . [by (4) of Art. 169] = O^ + OB2n ... (3) which proves the 248 PLANE TRIGONOMETRY. 171, Cotes Properties of the Circle. — These are particu-lar cases of De Moivres property ofthe circle. (1) Let OP, produced if necessary,meet the circle at A, and let AB = BC = CD, etc.,-*—; n then nO is a multiple of 2tt. Hence we have from (3) of Art. 170, after taking the square root of both members, OB* _ oP = PB • PC . PD ... to n factors ... I. (2) Let the arcs AB, BC, •••be bisected in the pointsa, b, •••; then we have, by (1), OB2* _ op2* = Pa • PB • P6 • PC • Pc ••• to 2n , by division, OBw + OF = Pa • Tb • Pc ... to n factors ... II. Cor. If the arcs AB, BC, ••• be trisected in the points &u &2> • ? •? then we haveOB^ + OF,-OG^+OP*i=== to 2n factors. 172. Resolve sin 6 into Factors. (1) Put x= 1; then we get from (3) of Art. 169 2(l-cos0)=2»(l-cos^l-^ -Yl-eoe-p + ^ (1) Put 6 = 2ji in (1), and let 2na = ir. 1 — cos 0 = 1 — cos
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