Expert: Sherman D. - 4/13/2008

Question
1. If a, b, c, and d are the four smallest positive angles, in ascending oder of magnitude, which have their sines equal to k, k>0, prove that
4sin a + 3sin(b/2) + 2 cos(c/2) + sin(d/2)=4k

2. If x=3sin(theta) - sin(3theta) and y=cos(3theta) + 3cos(theta), prove that x^(2/3) + y^(2/3) = 4^(2/3)  

Answer
x = 3sin(A) - sin(3A)
x = 3sin(A) - sin(2A + A)
x = 3sin(A) - (sin(2A)cos(A) + sin(A)cos(2A))
x = 3sin(A) - (2sin(A)cos(A)^2 + sin(A)(1 - 2sin(A)^2))
x = 3sin(A) - (2sin(A)(1 - sin(A)^2) + sin(A) - 2sin(A)^3)
x = 3sin(A) - (2sin(A) - 2sin(A)^3 + sin(A) - 2sin(A)^3)
x = 3sin(A) - (3sin(A) - 4sin(A)^3)
x = 3sin(A) - 3sin(A) + 4sin(A)^3
x = 4sin(A)^3

y = cos(3A) + 3cos(A)
y = cos(2A + A) + 3cos(A)
y = (cos(2A)cos(A) - sin(2A)sin(A)) + 3cos(A)
y = ((1 - 2sin(A)^2)cos(A) - 2cos(A)sin(A)^2) + 3cos(A)
y = cos(A) - 2cos(A)sin(A)^2 - 2sin(A)^2cos(A) + 3cos(A)
y = 4cos(A) - 4cos(A)sin(A)^2
y = 4cos(A)(1 - sin(A)^2)
y = 4cos(A)(cos(A)^2)
y = 4cos(A)^3

if you insert that into x^(2/3) + y^(2/3) = 4^(2/3)

(4sin(A)^3)^(2/3) + (4cos(A)^3)^(2/3) = 4^(2/3)
2sin(A)^2 + 2cos(A)^2 = 2
2(sin(A)^2 + cos(A)^2) = 2
2(1) = 2
2 = 2

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