Question

If cos-1x+cos-1y+cos-1z=pi and x+y+z=3/2, then what is the value of 2x+3y-5z?

Answer 1

Answer:

Let cos–1x = α, cos–1y = β and cos–1z = γ. Then cos α = x, cos β = y, cos γ = z

=> cos–1x + cos–1y + cos–1z = π

=> cos–1x + cos–1y = π - cos–1z

=> α + β = π - γ

=> cos(α + β) = cos(π - γ)

=> cosα. cosβ - sinα . sinβ = -cosγ

=> cosα. cosβ - √(1-cos2α) √(1-cos2β) = -cosγ => xy - √(1-x2) √(1-y2) = -z

=> xy + z = √(1-x2) √(1-y2)

=> (xy + z)2 = (1-x2)(1-y2)

=> x2 y2 + z2 + 2xyz = 1 – y2 – x2 + x2y2

=> x2 + y2 + z2 + 2xyz = 1