Sine theorem. pāʻoihana a triangle mauna

Ma ka like ana o triangle mauna involuntarily aia mea he ninau no helu ana i ka pilina ma waena o ko lākou mauʻaoʻao a me ka pana pua. I ka anahonua, ka theorem o cosines i hāʻawi mai a me sines ka loa piha pane i ka pilikia. I ka nui o nāʻano makemakika, ka 'aoʻao a me nā papakuhikuhi, kānāwai, theorems a me na rula i mea e okoa halawai lokahi, concise a me ka oluolu, e hanai i ka pio i loko o ia mau mea. Sine theorem mea he Kuhina Nui mamuli o ia ka makemakika hoʻomanaʻo. Inā ka waha 'ano, a he wahi mea i kekahi keʻakeʻa nei i loko o ka naauao o ka makemakika rula, ka wā e nānā aku i ka makemakika haʻilula a pau i koke mea hina i wahi.

Ka mua 'ike e pili ana i keia theorem i loaʻa i loko o ka palapala o ka hoike ana o ka mea, i loko o ka? Aieao o ka makemakika hana o Nasir AL-olowalu AL-Tusi, ka launa pū hoʻi i kaʻumikumamākolu kenekulia.

E hele kokoke i ka pilina ma waena o nāʻaoʻao a me ka pana pua i loko o kekahi triangle, e pono ke kūʻai 'ana i kumumanaʻo i ka sine theorem e leie aku mākou e hoʻoponopono i na mea he nui makemakika pilikia, a me ka anahonua o ke kānāwai loaʻa noi i loko o ka likeʻole ma o ka mea hiki kanaka haʻawina.

Ua sine theorem hākālia nō ia no kekahi triangle ua wehewehe aku ma ka proportionality aoao e ku pono ana i kihi o sines. Aia ka hoi o ka lua o ka hapa o keia theorem, e like me a ka lākiō o kekahiʻaoʻao o ka triangle e ku pono ana i ka sine o ka huina like, ua like me ke anawaena o ka p ÷ ai i ho'ākāka 'ia e pili ana i ka triangle ma lalo o noonoo.

Ma ka haʻilula kēia hōʻike nana e like

i / ma Sinai = b. / sinB = c. / sinC = 2R

Ua i hooiaio o ka theorem o sines, i loko o nā wale nō o ka heluna o nā puke i loaʻa i loko o ka waiwai likeʻole ma o wale nō.

No ka laʻana, noonoo kekahi o na hoike, e haawi ana i ka wehewehe ana o ka hapa mua o ka theorem. E hana i kēia, mākou e noi aku i ka hooiaio ana i ka hōʻike i ke aloha sinC = c. Sina.

Ma ka ākeʻakeʻa kumu triangle ABC,? Ieoaeunoai ke kiekie BH. I kekahiʻano, i ka? Ieoaeunoai H, e moe ana ma ka Hoʻohana AC, a me ka mea 'ē aʻe ma waho ka mea, ke kaumaha ma muli o ka nui o na pana pua ma ka vertices o na triangle mauna. Ma ka hihia mua, i ke kiekie hiki ke hoikeia ma ka pana pua, a aoao o ka triangle like BH = he sinC a me BH = pela aku ma Sinai, i ka ka mea i koi 'hoike.

I ka wa a ka H-wahi mea ma waho o ke Hoʻohana AC, ua hiki ke kiʻi i nā kēia pāʻoihana.

BH = He sinC a me VL c hewa (180-A) = pela aku ma Sinai;

a BH = ka hewa (180-C) = a sinC a me VL c Sina.

E like me 'oe ke ike, nānā' ole o ka manao koho, ua hiki i ka mea makemake hopena.

Ka hooiaio ana o ka lua o ka hapa o ka theorem, e hoopai mai, e wehewehe i ka pōʻai a puni ka triangle. Ma kekahi o ka triangle altitudes, no kekahi laʻana B, kükulu i ka hoa kanaka anawaena. Ke kūpono wahi ma luna o ka hoa kanaka D ua pili i kekahi o ke kiekie o ka triangle, e keia ia i ka wahi A o ka triangle.

Inā na triangle mauna loaa ABD a me ABC mākou noonoo, ua hiki ke ike i ka like o ka pana pua C me D (ka mea, i ka nānā 'ana ma luna o ka mea hookahi piʻo). A ua haawi aku i ka huina A, ua like me he kanaiwa degere o ka hewa D = pela aku / 2R, a me ka hewa C = pela aku / 2R, QED.

Sine theorem o ka ho omaka no ka laulā huahelu o kekahi mau hana. A mau attraction o kona pili ana, e like me ka corollary o Theorem mākou e hiki ke hai aku i ka waiwai o ka triangle aoao, kue pana pua, a me ka ke kahahńnai o (anawaena) o ka p ÷ ai circumscribed a puni ka triangle. Ka naau kupono, a loaʻa nui o ka haʻilula hōʻike no kēia kaha makemakikaʻa i, 'ae' ana i kaiāulu hoʻohana i kēia theorem, e hoʻoponopono i nā pilikia ma ke kumu o kekahi mau ÿano mea countable (kiʻiaka rula, papa, a pela aku.), Akā,ʻo ka hōʻeaʻana mai o ka hana kanaka ikaika Me na manao ona ua i kuʻu iho pilina o keia theorem.

Kēia theorem mea ole wale hapa o na koi papa o kiʻekiʻe kula anahonua, akā, hope hoʻohana 'ia ma kekahi mau hana me ka hana pono.