I-sine theorem. Ukuxazulula izintathu

Ukutadisha kwama-triangles kuphakamisa umbuzo wokubala ubudlelwane obuphakathi kwamacala nezinhlangothi zawo. E-geometry, i -cosine ne-sine theorem inikeza impendulo ephelele kunazo zonke yokuxazulula le nkinga. Ngezinkulumo eziningi zezibalo ezahlukene kanye namafomula, imithetho, imibono nemithetho, kunjalo ukuthi bahlukile ngokuvumelana okungajwayelekile, ngokushelela futhi okulula ekudluliseni incazelo equkethwe kuwo. I-sine theorem iyisibonelo esicacile sokusungulwa kwesibalo esinjalo. Uma kuhunyushwa ngomlomo kunesithiyo esithile ekuqondeni lo mthetho wezibalo, ke uma ubheka ifomati yamathemikhali yonke into ngokushesha ifika endaweni.

Ukwaziswa kokuqala ngalolu daba lwatholakala ngesimo sobufakazi bakhe esimweni somsebenzi wezibalo ka Nasir ad-Din Al-Tusi, wekhulu leshumi nantathu.

Ukusondela eduze kokucubungula kwesilinganiselo sesisindo futhi ubheke kunoma iyiphi inxantathu, kufanele kuqaphele ukuthi i-sine theorem ivumela ukuxazulula izinkinga eziningi zezibalo, kanti lo mthetho we-geometry uthola isicelo saso ezinhlobonhlobo zemisebenzi eyenziwa ngabantu.

I-sine theorem ngokwayo ithi ukuthi kunoma yimuphi unxantathu ukulinganisa kwezinhlangothi ezonweni zezingxube eziphikisanayo kuyisici. Kukhona nengxenye yesibili yale ngqungquthela, ngokusho ukuthi isilinganiso salo naluphi uhlangothi lwenxantathu kuze kube sine we-angle ehlukile lilingana nobubanzi bomjikelezo ochazwe eduze noxantathu ocatshangelwayo.

Ngendlela yefomula, le nkulumo ibukeka sengathi

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

Unomlando wokuqinisekisa, okuyizinguqulo ezihlukahlukene zezifundo ezinikezwe ngezinhlobo ezihlukahlukene zezinguqulo.

Isibonelo, cabanga ngenye yezinkomba ezichaza ingxenye yokuqala ye-theorem. Kuze kube sekugcineni, masibekele umgomo wokufakazela ukufaneleka kwenkulumo a SinC = C SinA.

Ku-triangle ye-ABC yesigcawu esingenalutho sakha ukuphakama kwe-BH. Ngenye yezinto ezahlukahlukene zokwakhiwa, H uzoqamba amanga kwi-AC, futhi kwenye ingaphezu kwemikhawulo yayo, kuye ngokuthi ama-angles atholakala kuma-triangles. Esikhathini sokuqala, ukuphakama kungabonakaliswa ngokwezingalo nezinhlangothi zontathu, njengeBH = sinC neBH = c sinA, okuyiwona ubufakazi obudingekayo.

Uma kwenzeka iphuzu H lingaphandle kwemingcele ye-AC, singathola izixazululo ezilandelayo:

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

Noma BH = isono (180-C) = sinC no BH = c sinA.

Njengoba ungabona, kungakhathaliseki ukuthi ukhetho lunikeza, siza emphumela oyifunayo.

Ubufakazi besigaba sesibili se-theorem sidinga ukuba sihlathulule umbuthano ojikeleze unxantathu. Ngenye yezingqimba zontathu, isibonelo B, sakha ububanzi bombuthano. Thola iphuzu kumbuthano D ngenye yokuphakama kontathu, ake kube yi-A yezinxantathu.

Uma sicabangela ukuxilongwa okubangelwa yi-ABD ne-ABC, khona-ke singabona ukulingana kwama-angles C no-D (asekelwe ku-arc eyodwa). Futhi ucabanga ukuthi i-angle A ingamazinga angamashumi ayisishiyagalolunye bese isona D = c / 2R, noma isono C = c / 2R, njengoba kudingeka.

I-sine theorem yindawo yokuqala yokuxazulula izinkinga ezahlukene ezahlukene. Ukukhanga okukhethekile kuyisisetshenziswa salo esiwusizo, ngenxa yombono we-theorem, sikwazi ukuchaza izindinganiso zamacala enxantathu, izingxenyana eziphikisanayo kanye nengqikithi (ububanzi) bendilinga elizungeza unxantathu. Ukulula nokufinyeleleka kwefomula echaza le nkulumo yamathekisthi yenza ukuthi kusetshenziswe kabanzi le ngqungquthela yokuxazulula izinkinga ngokusebenzisa amadivaysi amabala angama- mechanical (abakwa-logarithmic, amatafula, njll), kodwa ngisho nokufika kwamakhompyutha anamandla enkonzweni yendoda akunciphisanga ukubaluleka kwalesi sihloko.

Lezi zifundo azihlanganisiwe kuphela enkambweni eyimpoqo ye-geometry yesikole samabanga aphansi, kodwa isetshenziswe futhi emagatsheni athile omsebenzi osebenzayo.