= sin(3π/14) - sin(π/14) - sin(5π/14)
Using the sum-to-product formula for sine (sin(A) - sin(B) = 2cos((A+B)/2)sin((A-B)/2)), we can simplify the given expression:
= 2cos((3π/14 + π/14)/2)sin((3π/14 - π/14)/2) - sin(5π/14)= 2cos(2π/7)sin(π/7) - sin(5π/14)
Now, we can use the double angle formula for cosine (cos(2A) = 2*cos^2(A) - 1) to simplify further:
= 2(2cos^2(π/7) - 1)sin(π/7) - sin(5π/14)= 2(2(cos^2(π/7) - sin^2(π/7)) - 1)sin(π/7) - sin(5π/14)= 2(2(cos^2(π/7) - (1 - cos^2(π/7))) - 1)sin(π/7) - sin(5π/14)= 2(2(2cos^2(π/7) - 1) - 1)sin(π/7) - sin(5π/14)= 2(4cos^2(π/7) - 3)sin(π/7) - sin(5π/14)
We can simplify that further by using the double angle formula for cosine one more time, but the expression will be long and complex. The final value would be approximately -0.412.
= sin(3π/14) - sin(π/14) - sin(5π/14)
Using the sum-to-product formula for sine (sin(A) - sin(B) = 2cos((A+B)/2)sin((A-B)/2)), we can simplify the given expression:
= 2cos((3π/14 + π/14)/2)sin((3π/14 - π/14)/2) - sin(5π/14)
= 2cos(2π/7)sin(π/7) - sin(5π/14)
Now, we can use the double angle formula for cosine (cos(2A) = 2*cos^2(A) - 1) to simplify further:
= 2(2cos^2(π/7) - 1)sin(π/7) - sin(5π/14)
= 2(2(cos^2(π/7) - sin^2(π/7)) - 1)sin(π/7) - sin(5π/14)
= 2(2(cos^2(π/7) - (1 - cos^2(π/7))) - 1)sin(π/7) - sin(5π/14)
= 2(2(2cos^2(π/7) - 1) - 1)sin(π/7) - sin(5π/14)
= 2(4cos^2(π/7) - 3)sin(π/7) - sin(5π/14)
We can simplify that further by using the double angle formula for cosine one more time, but the expression will be long and complex. The final value would be approximately -0.412.