Ib ceg ntawm kev ua lej uas cov menyuam kawm ntawv muaj kev nyuaj siab tshaj plaws yog trigonometry. Tsis muaj qhov xav tsis thoob: txhawm rau ua kom muaj kev paub txog thaj chaw no, koj xav tau spatial xav, muaj peev xwm nrhiav tau sines, cosines, tangents, cotangents siv cov qauv, ua kom yooj yim cov kab lus, thiab siv tau tus lej pi hauv kev suav. Tsis tas li ntawd, koj yuav tsum muaj peev xwm siv tau trigonometry thaum ua pov thawj theorems, thiab qhov no yuav tsum muaj kev tsim lej cim xeeb lossis muaj peev xwm los txiav txim siab txoj hauv kev sib txuas.
Keeb Kwm ntawm Trigonometry
Kev taw qhia txog kev tshawb fawb no yuav tsum pib nrog lub ntsiab lus ntawm sine, cosine thiab tangent ntawm lub kaum sab xis, tab sis ua ntej koj yuav tsum paub seb qhov trigonometry ua li cas feem ntau.
Keeb kwm, txoj cai peb sab yog lub ntsiab ntawm kev tshawb fawb hauv seem ntawm kev tshawb fawb lej. Lub xub ntiag ntawm lub kaum sab xis ntawm 90 degrees ua rau nws muaj peev xwm ua tau ntau yam haujlwm uas tso cai rau obsab thiab ib lub ces kaum los yog ob lub ces kaum thiab ib sab los txiav txim qhov tseem ceeb ntawm tag nrho cov tsis muaj nyob rau hauv nqe lus nug. Yav dhau los, tib neeg pom tus qauv no thiab pib siv nws hauv kev tsim vaj tsev, kev taw qhia, hnub qub thiab txawm kos duab.
Inception
Thaum xub thawj, tib neeg tau tham txog kev sib raug zoo ntawm cov ces kaum thiab ob sab nkaus xwb ntawm qhov piv txwv ntawm txoj cai peb sab. Tom qab ntawd cov qauv tshwj xeeb tau pom, uas ua rau nws muaj peev xwm nthuav dav thaj tsam ntawm kev siv nyob rau hauv lub neej txhua hnub ntawm seem ntawm kev ua lej.
Kev kawm ntawm trigonometry hauv tsev kawm hnub no pib nrog cov duab peb sab, tom qab ntawd cov kev paub tau txais yog siv los ntawm cov menyuam kawm ntawv hauv physics thiab daws cov kev sib npaug trigonometric abstract, ua haujlwm nrog uas pib hauv high school.
Spherical trigonometry
Tom qab ntawd, thaum kev tshawb fawb mus txog qib tom ntej ntawm kev txhim kho, cov qauv nrog sine, cosine, tangent, cotangent pib siv rau hauv cov duab puab kheej kheej, qhov twg lwm txoj cai siv, thiab cov lej ntawm cov kaum hauv daim duab peb sab yog ib txwm muaj ntau dua. kub ntawm 180 degrees. Tshooj lus no tsis tau kawm hauv tsev kawm ntawv, tab sis nws yuav tsum paub txog nws lub neej, yam tsawg kawg yog vim lub ntiaj teb saum npoo av, thiab qhov chaw ntawm lwm lub ntiaj teb, yog convex, uas txhais tau hais tias txhua qhov cim ntawm qhov chaw yuav yog "arc-shaped. " nyob rau hauv peb-dimensional qhov chaw.
Tau lub ntiaj teb thiab ib txoj xov. Txuas cov xov rau ob lub ntsiab lus ntawm lub ntiaj teb kom nws taut. Them mloog - nws tau txais cov duab ntawm arc. Nws cuam tshuam nrog cov ntawv zoo li nospherical geometry siv nyob rau hauv geodesy, astronomy thiab lwm yam theoretical thiab siv teb.
daim duab peb sab
Tau kawm me ntsis txog txoj hauv kev ntawm kev siv trigonometry, cia peb rov qab mus rau qhov yooj yim trigonometry txhawm rau nkag siab ntxiv tias sine, cosine, tangent yog dab tsi, cov kev suav yuav ua tau nrog lawv cov kev pab thiab cov qauv siv dab tsi.
Ua ntej ntawm tag nrho, koj yuav tsum nkag siab cov ntsiab lus ntsig txog daim duab peb sab. Ua ntej, lub hypotenuse yog sab opposite lub kaum sab xis 90 degree. Nws yog tus ntev tshaj plaws. Peb nco ntsoov tias raws li Pythagorean theorem, nws tus lej tus nqi sib npaug ntawm cov hauv paus ntawm cov lej ntawm ob sab.
Piv txwv li, yog tias ob sab yog 3 thiab 4 centimeters, qhov ntev ntawm lub hypotenuse yuav yog 5 centimeters. Los ntawm txoj kev, cov neeg Iyiv thaum ub paub txog qhov no txog plaub thiab ib nrab txhiab xyoo dhau los.
Ob sab seem uas tsim lub kaum sab xis yog hu ua ob txhais ceg. Tsis tas li ntawd, peb yuav tsum nco ntsoov tias cov sum ntawm cov ces kaum nyob rau hauv ib daim duab peb sab nyob rau hauv ib tug rectangular coordinate system yog 180 degrees.
Definition
Thaum kawg, muaj kev nkag siab zoo ntawm lub hauv paus geometric, peb tuaj yeem tig mus rau lub ntsiab lus ntawm sine, cosine thiab tangent ntawm lub kaum sab xis.
Lub sine ntawm lub kaum sab xis yog qhov piv ntawm cov ceg sab nraud (uas yog, sab nraud ntawm lub kaum sab xis xav tau) rau lub hypotenuse. Lub cosine ntawm lub kaum sab xis yog qhov piv ntawm cov ceg uas nyob ib sab mus rau lub hypotenuse.
Nco ntsoov tias tsis yog sine lossis cosine tuaj yeem loj dua ib qho! Vim li cas?Vim tias lub hypotenuse yog los ntawm lub neej ntawd qhov ntev tshaj plaws ntawm ib daim duab peb sab. Txawm hais tias ntev npaum li cas ntawm txhais ceg, nws yuav luv dua li lub hypotenuse, uas txhais tau hais tias lawv qhov piv yuav ib txwm tsawg dua ib qho. Yog li, yog tias koj tau txais sine lossis cosine nrog tus nqi ntau dua 1 hauv cov lus teb rau qhov teeb meem, nrhiav qhov yuam kev hauv kev suav lossis kev xav. Cov lus teb no tsis yog lawm.
Thaum kawg, qhov tangent ntawm lub kaum sab xis yog qhov piv ntawm sab nraud mus rau sab uas nyob ib sab. Tib qhov tshwm sim yuav muab faib cov sine los ntawm cosine. Saib: Raws li cov qauv, peb faib qhov ntev ntawm sab los ntawm lub hypotenuse, tom qab ntawd peb faib los ntawm qhov ntev ntawm sab thib ob thiab muab faib los ntawm hypotenuse. Yog li, peb tau txais qhov sib piv ib yam li hauv lub ntsiab lus ntawm tangent.
Cotangent, feem, yog qhov piv ntawm sab uas nyob ib sab mus rau lub ces kaum mus rau sab nraud. Peb tau txais cov txiaj ntsig zoo ib yam los ntawm kev faib chav tsev los ntawm tangent.
Yog li, peb tau txiav txim siab cov ntsiab lus ntawm dab tsi yog sine, cosine, tangent thiab cotangent, thiab peb tuaj yeem cuam tshuam nrog cov qauv.
Cov qauv yooj yim
Hauv trigonometry ib qho tsis tuaj yeem ua yam tsis muaj qauv - yuav ua li cas thiaj nrhiav tau sine, cosine, tangent, cotangent yam tsis muaj lawv? Tab sis qhov no yog qhov yuav tsum tau ua thaum daws teeb meem.
Thawj tus qauv koj yuav tsum paub thaum pib kawm trigonometry hais tias cov lej ntawm cov squares ntawm lub sine thiab cosine ntawm lub kaum sab xis yog sib npaug rau ib qho. Cov qauv no yog qhov tshwm sim ncaj qha ntawm Pythagorean theorem, tab sis nws txuag lub sijhawm yog tias koj xav paub txog tus nqi ntawm lub kaum sab xis, tsis yog sab.
Ntau tus menyuam kawm ntawv tsis nco qab tus qauv thib ob, kuj heevnrov hauv kev daws teeb meem hauv tsev kawm ntawv: qhov sib npaug ntawm ib qho thiab lub xwmfab ntawm lub tangent ntawm lub kaum sab xis yog sib npaug rau ib qho muab faib los ntawm square ntawm lub cosine ntawm lub kaum sab xis. Ua tib zoo saib: tom qab tag nrho, qhov no yog tib nqe lus raws li hauv thawj cov qauv, tsuas yog ob sab ntawm tus kheej tau muab faib los ntawm square ntawm cosine. Nws hloov tawm tias kev ua lej yooj yim ua rau cov qauv trigonometric tsis paub meej. Nco ntsoov: paub tias lub sine, cosine, tangent thiab cotangent yog dab tsi, cov cai hloov dua siab tshiab thiab ob peb cov qauv yooj yim, koj tuaj yeem ua tau txhua lub sijhawm ntawm nws tus kheej kom tau txais cov qauv uas xav tau ntau dua ntawm ib daim ntawv.
ob lub kaum sab qauv qauv thiab ntxiv cov lus sib cav
Ob tug qauv ntxiv uas yuav tsum tau kawm yog muaj feem xyuam rau cov sine thiab cosine qhov tseem ceeb rau cov lej thiab qhov sib txawv ntawm cov ces kaum. Lawv muaj nyob rau hauv daim duab hauv qab no. Thov nco ntsoov tias nyob rau hauv thawj rooj plaub, lub sine thiab cosine yog multiplied ob lub sij hawm, thiab nyob rau hauv lub thib ob rooj plaub, lub pairwise khoom ntawm lub sine thiab cosine yog ntxiv.
Tseem muaj cov qauv txuam nrog ob lub kaum sab xis sib cav. Lawv tau tag nrho los ntawm cov yav dhau los - raws li kev xyaum, sim ua rau lawv tus kheej, noj lub kaum sab xis ntawm alpha sib npaug rau lub kaum sab xis ntawm beta.
Thaum kawg, nco ntsoov tias ob lub kaum sab xis qauv tuaj yeem hloov pauv kom txo qis qib ntawm sine, cosine, tangent alpha.
Theorems
Ob lub ntsiab lus tseem ceeb hauv trigonometry yog lub theorem sine thiab cosine theorem. Nrog kev pab los ntawm cov theorems, koj tuaj yeem nkag siab yooj yim yuav ua li cas thiaj nrhiav tau sine, cosine thiab tangent, thiab yog li thaj tsam ntawm daim duab, thiab qhov loj.txhua sab, thiab lwm yam.
Lub sine theorem hais tias vim kev faib qhov ntev ntawm txhua sab ntawm daim duab peb sab los ntawm tus nqi ntawm lub kaum sab xis, peb tau txais tib tus lej. Ntxiv mus, tus lej no yuav sib npaug rau ob lub voj voog ntawm lub voj voog ncig, piv txwv li lub voj voog uas muaj tag nrho cov ntsiab lus ntawm daim duab peb sab.
Lub theorem cosine generalizes Pythagorean theorem, projecting nws mus rau txhua daim duab peb sab. Nws hloov tawm hais tias los ntawm cov lej ntawm cov squares ntawm ob sab, rho tawm lawv cov khoom, muab faib los ntawm ob lub cosine ntawm lub kaum sab xis nyob ib sab ntawm lawv - qhov txiaj ntsig tshwm sim yuav sib npaug rau lub square ntawm peb sab. Yog li, Pythagorean theorem hloov mus ua ib rooj plaub tshwj xeeb ntawm cosine theorem.
Kev ua yuam kev vim tsis xav txog
Txawm tias paub tias sine, cosine thiab tangent yog dab tsi, nws yooj yim rau kev ua yuam kev vim tsis muaj lub siab xav lossis yuam kev hauv kev suav yooj yim. Kom tsis txhob yuam kev li no, cia peb mus saib cov uas nyiam tshaj plaws.
Ua ntej tshaj plaws, tsis txhob hloov cov feem ntau rau cov lej ua ntej tau txais qhov kawg tshwm sim - koj tuaj yeem tso cov lus teb raws li feem ntau, tshwj tsis yog hais tias lwm yam hauv qhov xwm txheej. Xws li kev hloov pauv tsis tuaj yeem hu ua yuam kev, tab sis nws yuav tsum nco ntsoov tias nyob rau txhua theem ntawm txoj haujlwm, cov hauv paus hniav tshiab tuaj yeem tshwm sim, uas, raws li tus sau lub tswv yim, yuav tsum raug txo kom tsawg. Hauv qhov no, koj yuav nkim sijhawm rau kev ua lej tsis tsim nyog. Qhov no yog qhov tseeb tshwj xeeb rau cov txiaj ntsig xws li lub hauv paus ntawm peb lossis ob, vim tias lawv tshwm sim hauv kev ua haujlwm ntawm txhua kauj ruam. Tib yam mus rau rounding."Ugly" tus lej.
Tom ntej, nco ntsoov tias cosine theorem siv rau txhua daim duab peb sab, tab sis tsis yog Pythagorean theorem! Yog hais tias koj yuam kev tsis nco qab rho ob zaug cov khoom ntawm ob sab multiplied los ntawm lub cosine ntawm lub kaum sab xis ntawm lawv, koj yuav tsis tau tsuas yog ib tug tsis ncaj ncees lawm tshwm sim, tab sis kuj yuav ua tau ib tug tag nrho misunderstandings ntawm cov kev kawm. Qhov no phem tshaj qhov ua yuam kev.
Thib peb, tsis txhob cuam tshuam qhov txiaj ntsig ntawm kaum ntawm 30 thiab 60 degrees rau sine, cosines, tangents, cotangents. Nco ntsoov cov txiaj ntsig no, vim tias sine ntawm 30 degrees yog sib npaug rau cosine ntawm 60, thiab rov ua dua. Nws yog ib qho yooj yim los sib tov lawv, thiab koj yuav inevitably tau ib tug yuam kev.
Daim ntawv thov
Ntau tus menyuam kawm ntawv tsis maj pib kawm trigonometry, vim lawv tsis nkag siab nws lub ntsiab lus siv. Dab tsi yog sine, cosine, tangent rau engineer lossis astronomer? Cov no yog cov tswv yim ua tsaug uas koj tuaj yeem xam qhov kev ncua deb mus rau cov hnub qub nyob deb, kwv yees lub caij nplooj zeeg ntawm lub meteorite, xa cov kev tshawb fawb mus rau lwm lub ntiaj teb. Yog tsis muaj lawv, nws yog tsis yooj yim sua los tsim ib lub tsev, tsim ib lub tsheb, xam lub load rau saum npoo los yog trajectory ntawm ib yam khoom. Thiab cov no tsuas yog cov piv txwv pom tseeb tshaj plaws! Tom qab tag nrho, trigonometry hauv ib daim ntawv lossis lwm qhov yog siv txhua qhov chaw, los ntawm suab paj nruag mus rau tshuaj.
In xaus
Yog li, koj paub dab tsi sine, cosine, tangent yog. Koj tuaj yeem siv lawv hauv kev suav thiab daws teeb meem hauv tsev kawm tiav.
Txhua lub ntsiab lustrigonometry raug txo qis rau qhov tseeb tias raws li paub tsis tau ntawm daim duab peb sab, nws yog ib qho tsim nyog los xam cov tsis paub. Muaj rau qhov tsis muaj nyob rau hauv tag nrho: qhov ntev ntawm peb sab thiab qhov loj ntawm peb lub kaum sab xis. Tag nrho qhov sib txawv ntawm cov dej num nyob hauv qhov tseeb tias cov ntaub ntawv sib txawv tau muab.
Yuav ua li cas thiaj nrhiav tau sine, cosine, tangent raws li paub ntev ntawm ob txhais ceg lossis hypotenuse, tam sim no koj paub. Txij li cov ntsiab lus no txhais tau hais tias tsis muaj dab tsi ntau dua li qhov piv, thiab qhov piv yog ib feem, lub hom phiaj tseem ceeb ntawm qhov teeb meem trigonometric yog nrhiav cov hauv paus hniav ntawm ib qho kev sib npaug sib npaug lossis ib qho kev sib npaug. Thiab ntawm no yog tsev kawm ntawv lej ib txwm yuav pab koj.
