148 CHAPTER 2. CURVES and then using the identities r 2 = x 2 + y 2 and y = r sin θ , we can write x 2 + y 2 = y which, after completing the square, can be rewritten as x 2 + p y − 1 2 P 2 = 1 4 . We recognize this as the equation of a circle centered at (0 , 1 2 ) with radius 1 2 . The polar equation r = sin 2 θ may appear to be an innocent variation on the preceding, but it turns out to be quite diFerent. Again the curve begins at the origin when θ = 0 and r increases with θ , but this time it reaches its maximum r = 1 when θ = π 4 , which is to say along the diagonal ( −→ p ( π 4 ) = ( 1 √ 2 , 1 √ 2 )), and then decreases, hitting r = 0 and hence the origin when θ = π 2 . Then r turns negative, which means that as θ goes from π 2 to π , the point −→ p ( θ ) lies in the fourth quadrant ( x > 0, y < 0); for π < θ < 3 π 2 , r is again positive, and the point makes a “loop” in the third quadrant, and ±nally for 3 π 2 < θ < 2 π , it traverses a loop in the second quadrant. After that, it traces out the same
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.