Engineering Calculus Notes 19

Engineering Calculus Notes 19 - is positive picks out the...

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1.1. LOCATING POINTS IN SPACE 7 Finally, we see that r = 0 precisely when P is the origin, so then the line is indeterminate: r = 0 together with any value of θ satis±es Equation ( 1.3 ), and gives the origin. For example, to ±nd the polar coordinates of the point P with rectangular coordinates ( 2 3 , 2), we ±rst note that r 2 = ( 2 3) 2 + (2) 2 = 16 . Using the positive solution of this r = 4 we have cos θ = 2 3 4 = 3 2 sin θ = 2 4 = 1 2 . The ±rst equation says that θ is, up to adding multiples of 2 π , one of θ = 5 π/ 6 or θ = 7 π/ 6, while the fact that sin
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Unformatted text preview: is positive picks out the rst value. So one set of polar coordinates for P is r = 4 = 5 6 + 2 n where n is any integer, while another set is r = 4 = p 5 6 + P + 2 n = 11 6 + 2 n. It may be more natural to write this last expression as = 6 + 2 n. For problems in space involving rotations (or rotational symmetry) about a single axis, a convenient coordinate system locates a point P relative to...
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