<prompt> Mlinganyo4 cos 27circ sqrta sqrtb sqrtc - sqrt dinashikilia kwa nambari kamili a, b, c , na d, ambapo b haigawanyiki kwa mraba wa prime, na d haiwezi kugawanywa kwa mraba wa prime. Pata a b c d.  <response>: Kwanza, tunapata thamani za cos 36circ. Let x cos 36circ na y cos 72circ. Kisha kwa fomula ya pembe mbili,y 2x2 - 1.Pia, cos 2 cdot 72circ cos 144circ -cos 36circ, hivyo-x 2y2 - 1.Tukiondoa milinganyo hii, tunapatax y 2x2 - 2y2 2x - yx y.Kwa kuwa x na y ni chanya, x y ni nonzero. Kwa hivyo, tunaweza kugawanya pande zote mbili kwa 2x y, kupatax - y frac12.Kisha y x - frac12. Inabadilisha kuwa y 2x2 - 1, tunapatax - frac12 2x2 - 1.Kisha 2x - 1 4x2 - 2, au 4x2 - 2x - 1 0. Kwa quadratic fomula,x frac1 pm sqrt54.Kwa kuwa x cos 36circ ni chanya, x frac1 sqrt5 4.Sasa,anzaaligncos 27circ sin 27circ2 cos2 27circ 2 cos 27circ dhambi 27circ sin2 27circ dhambi 54zingira 1 cos 36circ 1 frac1 sqrt54 1 frac5 sqrt54.endalignKwa vile cos 27circ sin 27circ ni chanya,cos 27circ sin 27circ fracsqrt5 sqrt52. quad quad 1Vile vile,anzaaligncos 27circ - sin 27circ2 cos2 27circ - 2 cos 27circ dhambi 27circ sin2 27circ -dhambi 54zingira 1 -cos 36circ 1 -frac1 sqrt54 1 frac3 - sqrt54.endalignKwa vile cos 27circ - sin 27circ ni chanya,cos 27circ - sin 27circ fracsqrt3 - sqrt52. quad quad 2Kuongeza milinganyo 1 na 2 na kuzidisha kwa 2, tunapata4 cos 27circ sqrt5 sqrt5 sqrt3 - sqrt5.Hivyo, a b c d 5 5 3 5 18. 