<prompt> Pata thamani ya juu zaidi yacos theta_1 dhambi theta_2 cos theta_2 dhambi theta_3 cos theta_3 dhambi theta_4 cos theta_4 dhambi theta_5 cos theta_5 dhambi theta_1, juu nambari zote halisi theta_1, theta_2, theta_3, theta_4, na theta_5.  <response>: Kwa Kutokuwa na Usawa kwa Kidogo, x - y2 ge 0 kwa nambari zote halisi x na y. Tunaweza kupanga hili upya kamaxy le fracx2 y22.Hii inaonekana kama AM-GM, lakini tunahitaji kuibainisha kwa nambari zote halisi, si nambari zisizo hasi tu.Kwa hivyo,anzaaligncos theta_1 dhambi theta_2 cos theta_2 dhambi theta_3 cos theta_3 dhambi theta_4 cos theta_4 dhambi theta_5 cos theta_5 dhambi theta_1 le fraccos2 theta_1 sin2 theta_22 fraccos2 theta_2 sin2 theta_32 quad fraccos2 theta_3 sin2 theta_42 fraccos2 theta_4 sin2 theta_52 fraccos 2 theta_5 sin2 theta_12 fraccos2 theta_1 sin2 theta_12 fraccos2 theta_2 sin2 theta_22 quad fraccos2 theta_3 sin2 theta_32 fraccos2 theta_4 sin2 theta_42 fraccos 2 theta_5 sin2 theta_52 frac52.endalignUsawa hutokea wakati theta_i zote ni sawa na 45circ, kwa hivyo thamani ya juu ni frac52. 