. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . os27r + g + lYn J\ n J ...( x2 - 2 a cos- J + 1 (2n-2)7r + (9 \a2-2 a cos- J + 1J— (3) (7o?\ Change a; into - in (3) and clear of fractions, anda ^.2n . f ~a o v we get #2n — 2 anaf cos 0 + a2n = ( ^ — 2 a# cos - + a2 .~(x2 - 2 ax cos ^+^ + a2Y^2 ~2ax cos i^+i + n A n J • • • to n factors (4) EXAMPLES. Find the quadratic factors of the following: 1. a8-2 a4 cos 60°+ 1 = 0. Ans. (x2 - 2x cos 15° + l)(ar - 2 x cos 105° + 1) X (x2 - 2 x cos 195° + 1) (x2 - 2 x cos 285° +
. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . os27r + g + lYn J\ n J ...( x2 - 2 a cos- J + 1 (2n-2)7r + (9 \a2-2 a cos- J + 1J— (3) (7o?\ Change a; into - in (3) and clear of fractions, anda ^.2n . f ~a o v we get #2n — 2 anaf cos 0 + a2n = ( ^ — 2 a# cos - + a2 .~(x2 - 2 ax cos ^+^ + a2Y^2 ~2ax cos i^+i + n A n J • • • to n factors (4) EXAMPLES. Find the quadratic factors of the following: 1. a8-2 a4 cos 60°+ 1 = 0. Ans. (x2 - 2x cos 15° + l)(ar - 2 x cos 105° + 1) X (x2 - 2 x cos 195° + 1) (x2 - 2 x cos 285° + 1) = 0. 2. a;10-2^cosl0o + l=0. Ans. (x2 - 2 x cos 2° + 1) (.t2 -2a? cos 74° + 1)X (a2 - 2 a? cos 146° + 1) (x2 - 2 x cos 218° + 1)(ar — 2 a? cos 2U0° + 1) = 0. BE MOIVBES PROPERTY OF THE CIRCLE. 247 »170. De Moivres Property of the Circle. — Let 0 be thecentre of a circle, P any point in itsplane. Divide the circumference inton equal parts BC, CD, DE,..., begin- (ning at any point B; and join 0 andP with the points of division B, C,D,... Let POB = 0; then willOB2n-20B • OP* cos n6l+OP2n. = 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 proposition.
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Keywords: ., bookcentury1900, bookdecade1900, booksubjecttrigono, bookyear1902
