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A62_soln - Math 237 Assignment 6 Solutions 1 Convert the...

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Math 237 Assignment 6 Solutions 1. Convert the following points from Cartesian coordinates to polar coordinates with 0 θ < 2 π . a) ( - 1 , - 3). Solution: We have r = radicalBig ( - 1) 2 + ( - 3) 2 = 2 and tan θ = - 3 - 1 = 3. Since the point is in quadrant 3 we get θ = 4 π 3 . Hence the point in polar coordinates is (2 , 4 π 3 ). b) (2 , 1). Solution: We have r = 2 2 + 1 2 = 5 and tan θ = 1 2 . Since the point is in quadrant 1 we get θ = arctan(1 / 2). Hence the point in polar coordinates is ( 5 , arctan(1 / 2)). 2. Convert the following points from polar coordinates to Cartesian coordinates. a) (3 , 2 π/ 3). Solution: We get x = 3 cos 2 π 3 = - 1 2 and y = 3 sin 2 π 3 = 3 3 2 , hence the point is parenleftBig - 1 2 , 3 3 2 parenrightBig in Cartesian coordinates. b) (2 , - π/ 6). Solution: We get x = 2 cos - π 6 = 3 and y = 2 sin - π 6 = - 1, hence the point is ( 3 , - 1) in Cartesian coordinates. 3. For each of the indicated regions in polar coordinates, sketch the region and find the area.
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