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Pre-Calc Exam Notes 149

Pre-Calc Exam Notes 149 - Polar Coordinates • Section 6.4...

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Unformatted text preview: Polar Coordinates • Section 6.4 149 x y O θ y x r ( r , θ ) ( x , y ) Figure 6.4.6 Figure 6.4.6 shows how to convert between polar coordinates and Cartesian coordinates. For a point with polar coordinates ( r , θ ) and Cartesian coordinates ( x , y ): Polar to Cartesian: x = r cos θ y = r sin θ (6.9) Cartesian to Polar: r = ± radicalBig x 2 + y 2 tan θ = y x if x negationslash= (6.10) Note that in formula (6.10), if x = 0 then θ = π /2 or θ = 3 π /2. Also, if x negationslash= 0 and y negationslash= 0 then the two possible solutions for θ in the equation tan θ = y x are in opposite quadrants (for ≤ θ < 2 π ). If the angle θ is in the same quadrant as the point ( x , y ), then r = radicalbig x 2 + y 2 (i.e. r is positive); otherwise r =− radicalbig x 2 + y 2 (i.e. r is negative). Example 6.15 Convert the following points from polar coordinates to Cartesian coordinates: (a) (2,30 ◦ ); (b) (3,3 π /4); (c) ( − 1,5 π /3) Solution: (a) Using formula (6.9) with r = 2 and...
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