Question

The positive integer value of `n >3` satisfying the equation `1/(sin(pi/n))=1/(sin((2pi)/n))+1/(sin((3pi)/n))i s`

Answer 1

Correct Answer - 7
Given, ` n gt3 in `Integer
` and (1)/(sin ((pi)/(n ))) = (1)/(sin ((2pi)/(n))) + (1)/(sin ((3pi)/(n)))`
` rArr (1)/(sin ""(pi)/(n)) - (1)/(sin ""(3pi)/(n)) = (1)/(sin ""(2pi)/(n)) `
`rArr ( sin ""( 3pi)/(n ) -sin ""( pi )/(n))/(sin ""(pi)/(n)* sin ""(3pi)/(n)) = (1)/(sin ""( 2pi )/(n))`
`rArr 2 cos (( 2pi )/( n )) * sin ""(pi)/(n) = ( sin ""(pi)/(n) * sin ""(3pi )/(n))/( sin ""( 2pi )/(n )) `
` rArr 2 sin ""(2pi)/(2)* cos ""( 2pi )/(n) = sin ""(3pi)/(n)`
`rArr " " sin ""(4pi)/(n) = sin ""( 3pi )/(n) `
` rArr " " ( 4pi)/(n ) = pi - ( 3pi )/(n)`
` rArr " " ( 7pi)/(n) = pi rArr n = 7`