<prompt> Acha ABCDEFG iwe heptagoni ya kawaida iliyo na kituo cha O. Acha M iwe kitovu cha pembetatu ABD. Tafuta cos2 pembe GOM.  <response>: Acha omega e2 pi i7. Kisha omega7 1, kwa hivyo omega7 - 1 0, ambayo huzingatia kamaomega - 1omega6 omega5 omega4 omega3 omega2 omega 1 0.Tangu omega neq 1, omega inatoshelezaomega6 omega5 omega4 omega3 omega2 omega 1 0.Tunaweka heptagoni ABCDEFG kwenye ndege, ili G iko 1, A iko omega, B iko omega2, na kadhalika.asyukubwa wa kitengo 2 cm;jozi A, B, C, D, E, F, G, M, O;G dir0;A dir3607;B dir23607;C dir33607;D dir 43607;E dir53607;F dir 63607;M A B D3;kuchoraA--B--C--D--E--F--G--mzunguko;kuchoraB--D--A;kuchoraM--O--G--mzunguko;lebo1, G, G;leboomega, A, A;leboomega2, B, B;leboomega3, C, C;leboomega4, D, D;leboomega5, E, E;leboomega6, F, F;nukta m, M, N;nukta0, 0,0, SW;asyKisha katikati ya pembetatu ABD ikom fracomega omega2 omega43.Sasa, kwa sheria ya kosine,cos angle GOM fracOG2 OM2 - GM22 cdot OG cdot OM.Tunaona kwamba OG 1, naanzaalignOM2 m2 m jumlam fracomega omega2 omega43 cdot frac1omega 1omega2 1omega43 fracomega omega2 omega4omega6 omega5 omega39 fracomega7 omega6 omega4 omega8 omega7 omega5 omega10 omega9 omega 79 frac1 omega6 omega4 omega 1 omega5 omega3 omega2 19 fracomega6 omega5 omega4 omega3 omega2 omega 39 frac29.mwishoalignPia,anzaalignGM2 1 - m2 1 - m1 - jumlam 1 - m - overlinem m overlinem 1 - fracomega omega2 omega43 - fracomega6 omega5 omega33 frac2 9 frac119 - fracomega6 omega5 omega4 omega3 omega2 omega3 frac119 frac13 frac149.endalignKisha OM sqrt23, hivyoanzaaligncos pembe GOM fracOG2 OM2 - GM22 cdot OG cdot OM frac1 29 - 1492 cdot 1 cdot sqrt23 frac-392 sqrt23 -frac12 sqrt2,mwishoalignambayo ina maanacos2 angle GOM left -frac12 sqrt2 kulia2 frac18. 