Sine theorem. tov ntawm cov vajvoos peb sab

Nyob rau hauv txoj kev tshawb no ntawm cov vajvoos peb sab involuntarily muaj ib lo lus nug ntawm kev xam cov kev sib raug zoo ntawm lawv sab thiab cov ces kaum. Nyob rau hauv geometry, cov theorem ntawm cosines thiab sines muab lub teb ntau lo lus teb rau qhov teeb meem. Lub abundance ntawm txawv zauv kab zauv thiab tus qauv, txoj cai, theorems thiab cov kev cai muaj xws li hais tias txawv txawv tshaj plaw kev sib raug zoo, cos thiab yooj yim rau pub ib tug neeg raug txim nyob rau hauv lawv. Sine theorem yog ib tug prime piv txwv ntawm xws li ib tug lej formulation. Yog hais tias cov lus hais txhais lus thiab tsis tau muaj ib tug tej yam khuam siab nyob rau hauv qhov kev nkag siab ntawm zauv cai, thaum koj saib ib tug zauv mis tag nrho ib zaug nws ntog rau hauv qhov chaw.

Tus thawj cov lus qhia txog qhov no theorem twb pom nyob rau hauv daim ntawv ntawm cov pov thawj ntawm nws nyob rau hauv lub moj khaum ntawm cov zauv ua hauj lwm ntawm Nasir al-Din al-Tusi, yos rov qab mus rau lub thib kaum peb caug xyoo.

Nce los ze zog mus rau lub kev sib raug zoo ntawm sab thiab cov ces kaum nyob rau hauv tej daim duab peb sab, yog tsim nyog sau cia hais tias lub sine theorem tso cai rau peb kom peb daws tau ntau zauv teeb meem, thiab cov geometry los ntawm txoj cai pom daim ntawv thov nyob rau hauv ib tug ntau yam ntawm cov tswv yim kev ua pej xeem.

Nws sine theorem hais tias rau tej daim duab peb sab yog yus muaj los ntawm proportionality sab opposite ntawm cov ces kaum sines. Muaj kuj yog ib tug thib ob yog ib feem ntawm no theorem, raws li uas tus piv ntawm tej sab ntawm lub duab peb sab opposite rau lub sine ntawm lub kaum sab xis yog sib npaug zos rau lub lub cheeb ntawm lub voj voog piav txog daim duab peb sab nyob rau hauv kev saib xyuas.

Nyob rau hauv ib lub mis no qhia zoo li

ib tug / sinA = b / sinB = c / sinC = 2R

Nws muaj pov thawj ntawm tus theorem ntawm sines, uas nyob rau hauv ntau yam versions ntawm cov phau ntawv muaj nyob rau hauv ib tug nplua nuj ntau yam ntawm versions.

Piv txwv li, xav txog ib yam ntawm cov proofs, muab ib tug piav qhia ntawm cov thawj ib feem ntawm lub theorem. Ua li no, peb yuav nug los ua pov thawj muab siab npuab lub qhia ib tug sinC = c sinA.

Nyob rau hauv ib tug arbitrary daim duab peb sab ABC, txua lub qhov siab BH. Nyob rau hauv ib embodiment, lub dlaim H yuav pw rau hauv lub ya AC, thiab lwm yam sab nraum nws, nyob ntawm seb qhov ntau ntawm cov ces kaum ntawm lub vertices ntawm lub duab peb ceg. Nyob rau hauv thawj cov ntaub ntawv, lub qhov siab yuav tsum los ntawm cov ces kaum thiab sab ntawm daim duab peb sab li BH = ib tug sinC thiab BH = c sinA, uas yog lub yuav tsum tau pov thawj.

Thaum lub H-point yog nyob sab nraum ntawm lub ya AC, peb yuav tau txais raws li nram no cov ntsiab:

BH = ib tug sinC thiab VL = c kev txhaum (180-A) = c sinA;

los yog BH = ib txoj kev txhaum (180-C) = thiab sinC thiab VL = c sinA.

Raws li koj tau pom, tsis hais txog ntawm kev tsim kev xaiv, peb txog ntawm tus yam tshwm sim.

Lub pov thawj ntawm lub thib ob yog ib feem ntawm lub theorem yuav yuav tsum tau rau peb kom peb piav qhia txog ib lub voj voog ib ncig ntawm lub daim duab peb sab. Los ntawm ib qho ntawm daim duab peb sab altitudes, piv txwv li B, txua ib lub voj voog txoj kab uas hla. Cov uas ua taw tes rau lub voj voog D yog kev cob cog rua rau ib qho ntawm ib qhov siab ntawm daim duab peb sab, cia qhov no yog point A ntawm daim duab peb sab.

Yog hais tias peb xav txog qhov tau cov vajvoos peb sab ABD thiab ABC, peb yuav saib tau lub koob pheej ntawm lawv ntawm cov ces kaum C thiab D (lawv yog raws li nyob rau hauv tib arc). Thiab muab hais tias lub kaum sab xis A yog sib npaug zos rau cuaj caum degrees kev txhaum D = c / 2R, los yog txoj kev txhaum C = c / 2R, QED.

Sine theorem yog starting taw tes rau ib tug ntau ntawm cov sib txawv raws qib. Ib qho attraction yog nws ua daim ntawv thov, raws li ib tug corollary ntawm Theorem peb muaj peev xwm mus txog tus nqi ntawm daim duab peb sab sab, tawm tsam cov ces kaum thiab lub voos kheej-kheej (inch) ntawm ib lub voj voog circumscribed nyob ib ncig ntawm lub daim duab peb sab. Lub simplicity thiab nyob rau ntawm mis piav txog qhov no xyuam xim kev hais lus, tau tso cai rau lug siv no theorem kom daws tau qhov teeb meem los ntawm txhais tau tias muaj ntau yam cov neeg kho tshuab pab kiag li lawm suav (swb cai, ntxhuav, thiab li ntawd.), Tab sis txawm lub sij hawm tuaj txog ntawm cov kev pab cuam neeg haib xam pab kiag li lawm yog tsis txo qis tseeb ntawm no theorem.

Qhov no theorem tsis yog tsuas yog ib feem ntawm qhov yuav tsum tau rau thaum kawg ntawm lub tsev kawm ntawv geometry, tab sis tom qab siv nyob rau hauv ib co industries xyaum.