Area bounded by curves y = [ cos A + cos B + cos C ] , y = [ 7 sin 2 A sin 2 B sin 2 C ] and curve ∣ x − 4∣ + ∣ y ∣ = 2 is - (where [.] denotes greatest integer function and A, B, C are vertices

Question

Area bounded by curves y = [ cos A + cos B + cos C ] , y = [ 7 sin 2 A ​ sin 2 B ​ sin 2 C ​ ] and curve ∣ x − 4∣ + ∣ y ∣ = 2 is - (where [.] denotes greatest integer function and A, B, C are vertices of triangle)

TestHub · Accepted Answer

Let S be the sum of the triangle's angle cosines and p the product of its half-angle sines. Then S=1+4p, with 0<p<=1/8. The floor values are 1 and 0. Inside the diamond, the width at height y is 4-2y.

$1 < S \leq 3/2, 0 < 7 p \leq 7/8. ⌊ S ⌋ = 1, ⌊ 7 p ⌋ = 0. A = \int_{0}^{1} (4 - 2 y) d y = [4 y - y^{2}]_{0}^{1} = 3.$

Mathematics - Area Under Curves Question with Solution | TestHub

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