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Question
Select the correct option from the given alternatives :
The value of `sin pi/14sin (3pi)/14sin (5pi)/14sin (7pi)/14sin (9pi)/14sin (11pi)/14sin (13pi)/14` is ....
Options
`1/16`
`1/64`
`1/128`
`1/256`
Solution
`1/64`
Explanation:
`sin pi/14sin (3pi)/14sin (5pi)/14sin (7pi)/14sin (9pi)/14sin (11pi)/14sin (13pi)/14`
= `sin pi/14sin (3pi)/14sin (5pi)/14 xx 1 xx sin(pi - (5pi)/14)sin(pi - (3pi)/14)sin(pi - pi/14) ...[because sin (7pi)/14 = sin pi/2 = 1]`
= `(sin pi/14sin (3pi)/14sin (5pi)/14)^2` ...[∵ sin(π – θ) = sin θ]
`sin pi/14sin (3pi)/14sin (5pi)/14`
= `sin(pi/2 - (3pi)/7)sin(pi/2 - (2pi)/7)sin(pi/2 - pi/7)`
= `cos (3pi)/7 cos (2pi)/7 cos pi/7`
= `1/(2sin(pi/7))[sin((2pi)/7)cos((2pi)/7)]cos (3pi)/7`
= `1/(4sin(pi/7))(sin (4pi)/7) cos(pi - (4pi)/7)`
= `- 1/(4sin(pi/7))(sin (4pi)/7 cos (4pi)/7)`
= `- 1/(8sin(pi/7)) sin((8pi)/7)`
= `-1/(8sin(pi/7)) (-sin(pi/7)) ...[sin((8pi)/7) = sin(pi + pi/7) = -sin(pi/7)]`
= `1/8`
∴ Required expression = `(1/8)^2 = 1/64`
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