Problem:
Let S be the set of all points (x,y) in the coordinate plane such that 0β€xβ€2Οβ and 0β€yβ€2Οβ. What is the area of the subset of S for which
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sin2xβsinxsiny+sin2yβ€43β?
Answer Choices:
A. 9Ο2β
B. 8Ο2β
C. 6Ο2β
D. 163Ο2β
E. 92Ο2β Solution:
For a fixed value of y, the values of sinx for which sin2xβsinxsiny+sin2y=43β can be determined by the quadratic formula. Namely,
Within S,sinx=sin(yβ3Οβ) implies x=yβ3Οβ. However, the case sinx=sin(y+3Οβ) implies x=y+3Οβ when yβ€6Οβ, and x=βy+32Οβ when yβ₯6Οβ. 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),(2Οβ,0),(0,2Οβ), and (2Οβ,2Οβ) shows that the inequality is true only in the shaded subregion. The area of this subregion is
