. Algebraic geometry; a new treatise on analytical conic sections . Fig. 48. PQ2 = 0P2 + 0Q2 - 20P. OQ cos POQ {a^ --= r^^ + r^^ -2r^r^cos{9^- 9^); + c3-2JccosA) PQ = Jr^^ + r^ - 2rjr^ cos (9^ - 9^). 10 POLAR CO-ORDINATES. [chap. 78. To find the area of the tricmgle formed by joining three pointswhose polar co-ordinates are given. Let P, Q, R be the pointswhose polar co-ordinatesare {ri,e^), {r^, 0^), (r^, 63)respectively. Join OP, OQ, OR. FlO. 49. A PQR = A OPQ + A OQR - A OPR = JOP. OQ sin POQ + JOQ. OR sin QOR - ^OP. OR sin POR = Ir^r^ sin (^^ - 6^) + ^r^r^ sin (9^ - 6^) - ^^Ya sin (^1
. Algebraic geometry; a new treatise on analytical conic sections . Fig. 48. PQ2 = 0P2 + 0Q2 - 20P. OQ cos POQ {a^ --= r^^ + r^^ -2r^r^cos{9^- 9^); + c3-2JccosA) PQ = Jr^^ + r^ - 2rjr^ cos (9^ - 9^). 10 POLAR CO-ORDINATES. [chap. 78. To find the area of the tricmgle formed by joining three pointswhose polar co-ordinates are given. Let P, Q, R be the pointswhose polar co-ordinatesare {ri,e^), {r^, 0^), (r^, 63)respectively. Join OP, OQ, OR. FlO. 49. A PQR = A OPQ + A OQR - A OPR = JOP. OQ sin POQ + JOQ. OR sin QOR - ^OP. OR sin POR = Ir^r^ sin (^^ - 6^) + ^r^r^ sin (9^ - 6^) - ^^Ya sin (^1 - ^3)= i [rf^ sin (01 - 0^ + r/3 sin {6^ - 0^) + r^r^ sin {0^ - \. 79. To find the equation of a straight line in polar co-ordinates. Let P(r, 6) be any pointon the straight line PN. Draw ON perpendicularto the line from the originO, and let ON =p, and makean /La with Ox, the initialline. From the aOPN,ON = OP cos PON, p = rcos(^-a),the equation required. Fio. 50. 80. By transforming the polar to rectangular co-ordinates, weshall see thatp = r cos (0-o.) must represent a straight line. The equation may be written p = r cos 8 eos a + r sin ^ sin a,
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