A6_soln - Math 237 Assignment 6 Due Friday July 4th 1...

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Math 237 Assignment 6 Due: Friday, July 4th 1. Convert the following points from Cartesian coordinates to polar coordinates with 0 θ < 2 π . a) (1 , - 1) Solution: We have r = radicalbig (1) 2 + ( - 1) 2 = 2 and tan θ = - 1 1 . Since the point is in the fourth quadrant we get θ = 7 π 4 hence we have the point ( 2 , 7 π 4 ). b) ( - 1 , 3) Solution: We have r = radicalBig ( - 1) 2 + ( 3) 2 = 2 and tan θ = 3 - 1 . Since the point is in second quadrant we get θ = 2 π 3 and hence have the point (2 , 2 π 3 ). 2. Convert the following points from polar coordinates to Cartesian coordinates. a) (2 , π/ 4) Solution: We have x = r cos θ = 2 cos π 4 = 2 and y = r sin θ = 2 sin π 4 = 2. Hence, we have the point ( 2 , 2). b) (6 , - π/ 6) Solution: We have x = r cos θ = 6 cos( - π 6 ) = 3 3 and y = r sin θ = 6 sin( - π 6 ) = - 3. Hence, we have the point (3 3 , - 3). 3. For each of the indicated regions in polar coords, sketch the region and find the area. a) The region enclosed by r = sin 3 θ .
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