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 ______.

पर्याय

• `1/16`

• `1/64`

• `1/128`

• `1/256`

Answer 1

The value of `sin  pi/14sin  (3pi)/14sin  (5pi)/14sin  (7pi)/14sin  (9pi)/14sin  (11pi)/14sin  (13pi)/14` is `\underline(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`