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

Here, `" "k=sin((pi)/(18))sin((5pi)/(18))sin((7pi)/(18))`
`" "=sin10^(@)sin50^(@)sin70^(@)`
`" "=sin10^(@)cos40^(@)*cos20^(@)`
`" "=(1)/(2)sin10^(@)[2cos40^(@)*cos20^(@)]`
`" "=(1)/(2)sin10^(@)[cos60^(@)+cos20^(@)]" "[because 2cosx*cosy=cos(x+y)+cos(x-y)]`
`" "=(1)/(2)sin10^(@)*(1)/(2)+(1)/(2)sin10^(@)cos20^(@)`
`" "=(1)/(4)sin10^(@)+(1)/(4)[sin30^(@)-sin10^(@)]`
`=(1)/(8)`

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If F(k)=(1+sin""(pi)/(2k))(1+sin(k-1)(pi)/(2k))(1+sin(2k+1)(pi)/(2k)),(1+sin(3k-1)(pi)/(2k)) then the value of F(1)+F(2)+F(3) is equal to

`(3)/(16)`

`(1)/(4)`

`(5)/(16)`

`(7)/(16)`

The value of sin"" (pi )/( 18) sin"" ( 5pi )/(18) sin"" ( 7pi )/( 18) is

`1//2`

`1//4`

`1//8`

`1//16`

sin^(2)pi/8+sin^(2)(3pi)/(8)+sin^(2)(5pi)/(8)+sin^(2)(7pi)/(8)=?

`-1`

`-2`