Problem:
Let S be the set of all points (x,y) in the coordinate plane such that 0≤x≤2π and 0≤y≤2π. What is the area of the subset of S for which
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sin2x−sinxsiny+sin2y≤43?
Answer Choices:
A. 9π2
B. 8π2
C. 6π2
D. 163π2
E. 92π2 Solution:
For a fixed value of y, the values of sinx for which sin2x−sinxsiny+sin2y=43 can be determined by the quadratic formula. Namely,
sinx=2siny±sin2y−4(sin2y−43)=21siny±23cosy
Because cos(3π)=21 and sin(3π)=23, this implies that
sinx=cos(3π)siny±sin(3π)cosy=sin(y±3π)
Within S,sinx=sin(y−3π) implies x=y−3π. However, the case sinx=sin(y+3π) implies x=y+3π when y≤6π, and x=−y+32π when y≥6π. Those three lines divide the region S into four subregions, within each of which the truth value of the inequality is constant. Testing the points (0,0),(2π,0),(0,2π), and (2π,2π) shows that the inequality is true only in the shaded subregion. The area of this subregion is
