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Engineering Calculus Notes 19

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

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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 satisfies Equation ( 1.3 ), and gives the origin. For example, to find the polar coordinates of the point P with rectangular coordinates ( 2 3 , 2), we first 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 first 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 θ
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Unformatted text preview: θ is positive picks out the ±rst value. So one set of polar coordinates for P is r = 4 θ = 5 π 6 + 2 nπ where n is any integer, while another set is r = − 4 θ = p 5 π 6 + π P + 2 nπ = 11 π 6 + 2 nπ. It may be more natural to write this last expression as θ = − π 6 + 2 nπ. 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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