I-sore theorem. Ukulungisa izilwanyana ezintathu

Ukufundwa kwamangxantathu kukuphakamisa umbuzo wokubala ubuhlobo phakathi kwamacandelo kunye nama-angles. Kwijometri, i-cosine kunye ne-sore theorem inika impendulo epheleleyo ekuxazululeni le ngxaki. Kwiintlobo ezininzi zeemathematika ezihlukeneyo kunye namafomula, imithetho, i-theorems kunye nemithetho, kukho into yokuba ihluke ngokuvisisana ngokungaqhelekanga, ngokucacileyo nangokulula ukuhambisa intsingiselo ebhalwe kuyo. I-theorem yesistim ngumzekelo ocacileyo wokuqulunqwa kwemathematika okunjalo. Ukuba kukho ukuchazwa ngamagama kukho nkontsho ethile ekuqondeni lo mgaqo weemathematika, ngoko ke xa ukhangela i-formula yemathematika yonke into ngokukhawuleza iwela endaweni.

Ingcaciso yokuqala malunga nale mfundiso ifunyenwe ngohlobo lobubungqina besikhokelo somsebenzi weMathem ad-Din Al-Tusi, ngenkulungwane yeshumi elinesithathu.

Ukusondela kufuphi nokuqwalaselwa komlinganiselo kunye nomngxuma kunoma yimuphi unxantathu, kuyafaneleka ukuba uqaphele ukuba i-theorem ye-sine ivumela ukuxazulula iingxaki ezininzi zeemathematika, ngelixa lo mthetho we-geometry ufumana ukusetyenziswa kwawo kwiintlobo ezahlukeneyo zokwenza abantu.

I-sore theorem ngokwayo ithi ukuba naluphi na unxantathu ukulingana kwamacala kwisono se-angles echaseneyo kuyimpawu. Kukho inxalenye yesibini yale ngqungquthela, ngokubhekiselele ukuba umlinganiselo waluphi na uhlangothi lwendangxantathu ukuya kwisinekile yecala elichaseneyo lilingana nobubanzi besangqa esichazwe ngasentla kunxantathu ecingelwayo.

Ngendlela yohlobo, eli binzana libukeka ngathi

A / sinA = b / sinB = c / sinC = 2R

Unomxholo wobungqina bendalo, efumaneka kwiinguqu ezahlukeneyo zeencwadi zezifundo kwiintlobo ezahlukeneyo zeenguqulelo.

Ngokomzekelo, cinga enye yobungqina obucacisa inxalenye yokuqala ye-theorem. Kule nto, masibekele injongo yokubonisa ubungqina begama elithi a SinC = C SinA.

Kwi-Triangle ye-triangle ye-ABC sinokwakha ubude bH. Ngenye yezinto ezahlukahlukeneyo zokwakha, H iya kulala kwicandelo le-AC, kwaye kwelinye elingaphaya kwemida yalo, kuxhomekeka kwii-angles kwii-vertices ze-triangles. Kwimeko yokuqala, ukuphakama kungabonakaliswa ngokweemeko kunye namacala omnxantathu, njenge-BH = sinC ne-BH = c sinA, obungqina obufunekayo.

Kwimeko apho iphuzu H lingaphandle kwemida yeC AC, sinokufumana izicwangciso ezilandelayo:

BH = sinC kunye neBH = c isono (180-A) = c sinA;

Okanye iBH = isono (180-C) = isonoC neBH = c sinA.

Njengoko unako ukubona, kungakhathaliseki ukuba kukhethwa ukwakhiwa, siza kwisiphumo esifunileyo.

Ububungqina benkqutyana yesibini ye-theorem sifuna ukuba sikwazi ukuchaza isangqa malunga nxantathu. Ngomnye weentsika zexantathu, umzekelo B, sakha ububanzi besangqa. Fumana ingqungquthela kwisangqa D kunye nenye yokuphakama kwengxantathu, masibe ngu-A kwinqatyana.

Ukuba sicinga i-triangles ezibangelwa yi-ABD kunye ne-ABC, ngoko siyakwazi ukubona ukulingana kwama-C kunye no-D (basusela kwi-arc enye). Xa sicinga ukuba i-angle A iinqununu ezingamashumi ayisithoba kwaye yenza isono D = c / 2R, okanye isono C = c / 2R, njengoko kufuneka.

I-theorem ye-sine yinqanaba lokusombulula uluhlu lweengxaki ezahlukeneyo. Ukukhangela okukhethekileyo kuyisicelo esisebenzayo, ngenxa ye-theorem, siyakwazi ukuxelisa ixabiso leenxalenye zomxantathu, ii-angles ezichaseneyo kunye ne-radius (ububanzi) besangqa sithatyathele unxantathu. Ukulula kunye nokufikeleleka kwefomula echaza le ncazelo yeemathematika yenza ukuba kube lula ukusebenzisa le ngqungquthela yokuxazulula iingxaki ngokusebenzisa amadivaysi ahlukeneyo okubala (i-logarithmic, iithebula, njl.), Kodwa kwanokufika kwamakhompyutha anamandla enkonzweni yendoda ayizange iyanciphise ukufaneleka kwalo mbhalo.

Le ngqungquthela ayifakiwe kuphela kwikhosi eyanyanzeliswayo yeJometri yesikolo sasekondari, kodwa iphinda isetyenziswe nakwamasebe athile okwenza umsebenzi.