<prompt> Amua thamani halisi yasqrtleft 2 - sin2 fracpi7 kulia kushoto 2 - sin2 frac2 pi7 kulia kushoto 2 - sin2 frac3 pi7 kulia.  <response>: Kwa ujumla, Na nadharia ya DeMoivre,anzaalignjina la operetacis n theta jina la operetacis thetan cos theta i sin thetan cosn theta binomn1 i cosn - 1 theta sin theta - binomn2 cosn - 2 theta sin2 theta - binomn3 i cosn - 3 theta sin3 theta dotsb.endalignKulingana na sehemu halisi na za kufikirika, tunapataanzaaligncos n theta cosn theta - binomn2 cosn - 2 theta sin2 theta binomn4 cos n - 4 theta sin4 theta - dotsb, sin n theta binomn1 cosn - 1 theta sin theta - binomn3 cosn - 3 theta sin 3 theta binomn5 cosn - 5 theta sin5 theta - dotsb.endalignKwa n 7,anzaaligndhambi 7 theta 7 cos6 theta sin theta - 35 cos4 theta sin3 theta 21 cos2 theta sin5 theta - dhambi 7 7 1 - sin2 theta3 dhambi theta - 35 1 - sin2 theta2 sin3 theta 21 1 - sin2 theta sin5 theta - sin7 theta -64 sin7 theta 112 dhambi5 theta - 56 sin3 theta 7 dhambi theta -dhambi theta 64 sin6 theta - 112 sin4 theta 56 sin2 theta - 7.endalignKwa theta frack pi7, k 1, 2, na 3, sin 7 theta 0, hivyo sin2 fracpi7, sin2 frac2 pi7, na sin2 frac3 pi7 ndio mizizi ya64x3 - 112x2 56x - 7 0.Hivyo,64 kushoto x - sin2 fracpi7 kulia kushoto x - sin2 frac2 pi7 kulia kushoto x - sin2 frac3 pi7 kulia 64x3 - 112x2 56x - 7kwa x. zote Tukichukua x 2, tunapata64 kushoto 2 - sin2 fracpi7 kulia kushoto 2 - sin2 frac2 pi7 kulia kushoto 2 - sin2 frac3 pi7 kulia 169,hivyosqrtleft 2 - sin2 fracpi7 kulia kushoto 2 - sin2 frac2 pi7 kulia kushoto 2 - sin2 frac3 pi7 kulia frac138. 