By Prof. H. Lennart Pearson (auth.), Carl E. Pearson (eds.)
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Extra info for Handbook of Applied Mathematics: Selected Results and Methods
Cos 4A = cos 4 A - 6 cos 2 A sin 2 A + sin4 A sin 4A =4 sin A cos3 A - 4 sin 3 A cos A and conversely =cos 4A + 6 cos 2 A cos4 A sin 2 A - sin4 A = cos 4A + 6 cos 2 A (1 - cos 2 A) - (1 - cos 2 A)2 8 cos4 A = cos 4A + 8 cos 2 A-I = cos 4A + 4(cos 2A + 1) - 1 Thus 8 cos4 A = cos 4A + 4 cos 2A + 3 and similarly, 8 sin4 A n L =cos 4A - C(n, k) cos kx = 2n 4 cos 2A + 3 cosn k=l n " ~ C(n, k) sin kx =2n cosn k=l X nx X nx 2 2 "2 cos 2"" - 1 (Ll 0-17) - sin- Also, for any real 0: and any real x =1= 2m 1T (m an integer) n L cos (0: + kx) = sin!
X Fig. 11·2 Polar coordina tes. 1 Angle of Inclination, Slope, Intercepts The angle of inclination of a line L is the smallest angle obtained when the + x·axis is rotated counterclockwise into coincidence with the line. Thus 0 ,;;;; a < 1800 • The slope of L is defined to be tan a and will be denoted by s. If PI (XI, YI), P2 (X2, Y2) are two points on L, then Y2 - YI X2 - XI s=--- r ---------'~"'-=+_----------II. 11·3 Intercept of a line on x· and y·axes. 11·6) Formulas from Algebra, Trigonometry and Analytic Geometry 43 The x-coordinate of the point where L intersects the x-axis is the x-intercept of L and similarly the y-coordinate of the point where L intersects the y-axis is the y-intercept of L.
11-1. To each point in the plane there corresponds an ordered pair of numbers, the x- and y-coordinates of the point, where the x-coordinate (abscissa) is the directed distance of the point from the y-axis and the y-coordinate (ordinate) is the directed distance of the point from the x-axis. The direction of measurement is taken to be positive for distance Formulas from A Igebra, Trigonometry and Analytic Geometry 41 y II X' ------0+-----0- X IV III Fig. 11-1 Cartesian coordinate system. measured horizontally to the right or vertically upwards, the opposites are taken to be negative.
Handbook of Applied Mathematics: Selected Results and Methods by Prof. H. Lennart Pearson (auth.), Carl E. Pearson (eds.)