By W J. Johnston
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Extra resources for An Elementary Treatise on Analytical Geometry, with Numerous Examples
E. measured round in the positive It will direction from § 57. With OX. this understanding, it will be found on examination that whatever be the position of the line the proofs in Arts. 54 and 55 are perfectly general, and that in all cases the equation is x The cos OC + y sin OC = p. generality of the proof in Art. 55 depends on this, that whatever be the Geometry Analytical 38 magnitude of sin OC OC, OA and cos OC have have always the same L58- OB and If in fig' the Eu- always the same sign, and also sign.
Ans. Ex. is > 3j* in the figure. Plot 4. A C \ TT the point (—3, if). in the figure. 5. Plot (3, 2 mr+ —J , where n is any integer. Ans. B in the figure. § 39- To transform from rectangular vice versa. ] §40. To express ^3 two points in terms of the distance between their polar co-ordinates. P (»\0i) Q (noi) Let the points be PQ = OP + OQ - 2 OP. -. 2 x xx rx cos 6 X P = y2 = ) PQ2 = (Xl = (rt - x2 ) 2 + Yl - y2 )2 cos Bt — r2 cos 2 ) 2 + = r^ (cos2 X + sin 2 X) + (rx sin X r2 2 (cos2 — r*! r2 cos r2 sin + r2 2 - 2 rx r2 cos (0 X - r2 sin + 2 2 = yx ) and of Q, 2 ) 2 ) (x2 y2 ).
Then — xx = y2 - yi = x2 Yi rx cos X x3 \ ri sin 0i J y3 x. — - xx yi = = r2 cos 2 r2 sin 2 = (x2 - xx) (y3 - y x ) - (x3 - xO (y2 - y = rx cos X r2 sin 2 — r2 cos 2 rx sin X = - rjrasinC^ - 2 ). Thus . §40, This clockwise = OPQ = \ OP. OQ sin POQ = \r in the order in a2 . x) is Co-ordinates <4-] § 4-2. The expression at the beginning of the last area OPQ = %rx r2 OPQ = |(rj sin Q area gives 25 = K x2 x r2 cos . yi-x 1 — sin (Q x 2 §, 2) r2 cos is positive if the order of (o, o), (x1 S 4-3. To Fig.
An Elementary Treatise on Analytical Geometry, with Numerous Examples by W J. Johnston