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If K=sin(pi/(18))sin((5pi)/(18))sin((7pi...

If `K=sin(pi/(18))sin((5pi)/(18))sin((7pi)/(18)),` then the numerical value of `K` is_____

Text Solution

Verified by Experts

The correct Answer is:
`(1)/(8)`
Using the relation,
`sin theta sin ((pi)/(3) - theta)sin ((pi)/(3) + theta)=(sin 3 theta)/(4)`
Taking `theta=(pi)/(18)`, we get
`sin .(pi)/(8).sin.(5pi)/(18).sin.(7pi)/(18)=(sin.(pi)/(6))/(4)=(1)/(8)`
Alternative Method
Given, `k=sin10^(@).sin50^(@).sin70^(@)`
`=cos80^(@).cos40^(@).cos20^(@)`
`=cos A . cos 2A. cos2^(2)A = (sin2^(3)A)/(2^(3)sinA)`
where, A = `20^(@)`
`=(sin160^(@))/(8sin20^(@))=(sin(180^(@)-20^(@)))/(8sin20^(@))=(sin20^(@))/(8sin20^(@))=(1)/(8)`
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