α、β、γ用a、b、c表示：

3個方向餘弦滿足：cosa^2+cosb^2+cosc^2=1

即：sina^2+sinb^2+sinc^2=2

1/sina^2+4/sinb^2+9/sinc^2

=(1/2)*(2/sina^2)+2*2/sinb^2+(9/2)*(2/sinc^2)

=(1/2)*(1+sinb^2/sina^2+sinc^2/sina^2)+2*(sina^2/sinb^2+1+sinc^2/sinb^2)

+(9/2)*(sina^2/sinc^2+sinb^2/sinc^2+1)

=(1/2+2+9/2)+sinb^2/(2sina^2)+sinc^2/(2sina^2)

+2sina^2/sinb^2+2sinc^2/sinb^2+9sina^2/(2sinc^2)+9sinb^2/(2sinc^2)

≥7+2(sinb*√2sina/(√2sina*sinb))+2(3sinc*sina/(√2sina*√2sinc))

+2(√2sinc*3sinb/(sinb*√2sinc))=18

等號成立的條件：sinb^2=2sina^2，sinc^2=3sina^2，2sinc^2=3sinb^2

但這個等號成立的條件比較苛刻，還要考慮sina^2+sinb^2+sinc^2=2的條件

即：6sina^2=2，即：sina^2=1/3，sinb^2=2/3，sinc^2=1

即：sina=√3/3，sinb=√6/3，sinc=1

此時1/sina^2+4/sinb^2+9/sinc^2取得最小值：18