Question

Without using tables, prove that `(sin12^0)(sin48^0)(sin54^0)=1/8`

Answer 1

`sin12^(@)sin 48^(@)sin 54^(@) = (1)/(2) (2sin 12^(@) sin 48^(@))sin 54^(@)`
`=(1)/(2)[cos(36^(@))-cos(60^(@))]sin54^(@)`
`=(1)/(2)(cos36^(@)-(1)/(2))sin54^(@)`
`=(1)/(4)(2cos 36^(@)sin 54^(@)-sin 54^(@))`
`=(1)/(4)(sin90^(@)+sin18^(@)-sin54^(@))`
`=(1)/(4) (1+(sqrt(5)-1)/(5)-(sqrt(5)+1)/(4))`
`=(1)/(4)(1+(sqrt(5)-1-sqrt(5)-1)/(4))`
`=(1)/(4)(1-(1)/(2))=(1)/(8)`