First slide

Introduction to 3-D Geometry

Question

If a line makes angles $α, β, γ$ with positive axes, the the range of $\sin α s i n β + \sin β s i n γ + \sin γ s i n α$ is

Moderate

A

$(\frac{- 1}{2}, 1)$
B

$(\frac{- 1}{2}, 2)$
C

(-1,2)
D

$(- 1, 2]$

Solution

${(\sin α + \sin β + s i n γ)}^{2} > 0$
$\sin^{2} α + \sin^{2} β + \sin^{2} γ - \sin α \sin β$
$- \sin β \sin γ - \sin γ \sin α \geq 0$
But $\sin α, \sin β, \sin γ > 0$

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