<prompt> Acha S iwe seti ya alama zote x,y katika ndege ya kuratibu ili 0leq xleq fracpi2 na 0leq yleqfracpi2. Ni eneo gani la kitengo kidogo cha S ambacho sin2 x -sin x sin y sin2 y leqfrac34,?  <response>: Kwa thamani isiyobadilika ya y, thamani za sin x ambazo sin2 x-sin xsin ysin2 yfrac34 zinaweza kubainishwa. kwa formula ya quadratic. Yaani, sin xfracsin ypmsqrtsin2 y-4sin2 y-frac342frac12sin ypmfracsqrt32cos y.Kwa sababu cos displaystyleleftfracpi3displaystyleright frac12 na sin displaystyleleftfracpi 3displaystyleright fracsqrt32, hii ina maana kwamba sin xcosdisplaystyleleftfracpi3displaystylerightsin ypmsin displaystyleleftfracpi3displaystyleright cos ysindisplaystyleleftypmfracpi3displaystylekulia.Ndani ya S, sin xsiny-fracpi3 ina maana xy-fracpi3. Walakini, kesi sinxsinyfracpi3 ina maana xyfracpi3 wakati yleqfracpi6, na x-yfrac2pi3 wakati ygeqfracpi6. Mistari hiyo mitatu inagawanya eneo S katika kanda nne, ndani ya kila moja ambayo thamani ya ukweli ya ukosefu wa usawa haibadilika. Kujaribu pointi 0,0, fracpi2,0, 0,fracpi2, na frac pi2,fracpi2 inaonyesha kuwa ukosefu wa usawa ni kweli katika eneo lenye kivuli pekee. Eneo la kanda hii ni displaystyleleftfracpi2displaystyleright2-frac12cdotdisplaystyleleftfracpi3displaystyleright 2-2cdotfrac12cdotdisplaystyleleftfracpi6displaystyleright2fracpi26. asyukubwa wa kitengo cm 3;chora0,0--1,0--1,1--0,1--mzunguko,dashed;jaza0,0.66--0.33,1--1,0.33--0.66,0--0,0--mzunguko,kijivu0.7;nukta 0,0;nukta 0,1;nukta 1,1;nukta 1,0;nukta 0.66,0;nukta 0,0.66;nukta 0.33,1;nukta 1,0.33;chora0,0.66--0.33,1--1,0.33--0.66,0,upana wa mstari0.7;lebo0,0,0,0,W;lebo0,fracpi2,0,1,W;lebofracpi2,0,1,0,E;lebofracpi2, fracpi2,1,1,E;chora1.1,0.43--0.56,-0.1,upana wa mstari0.7;chora1.1,0.23--0.23,1.1,upana wa mstari0.7;chora-0.1,0.56--0.43,1.1,upana wa mstari0.7;leboxyfracpi3,1.1,0.43,E;leboxy-fracpi3,0.43,1.1,NE;lebox-yfrac2pi3,0.23,1.1,NW;asy 