Txhua daim duab peb sab muaj ib lub kaum sab xis (90 degrees), thiab qhov hypotenuse yog ib sab uas tsis sib thooj. Qhov hypotenuse yog qhov ntev tshaj plaws ntawm daim duab peb sab, thiab nws kuj yooj yim heev los nrhiav nws siv ob peb txoj hauv kev sib txawv. Kab lus no yuav qhia koj yuav nrhiav qhov ntev ntawm qhov hypotenuse siv Pythagorean theorem yog koj paub qhov ntev ntawm ob sab ntawm daim duab peb sab. Tom ntej no, kab lus no yuav qhia koj li cas txhawm rau txheeb xyuas qhov hypotenuse ntawm qee qhov tshwj xeeb daim duab peb sab uas tshwm sim ntau zaus hauv kev xeem. Thaum kawg, tsab xov xwm no yuav qhia koj yuav nrhiav qhov ntev ntawm qhov hypotenuse siv Txoj Cai ntawm Sine yog tias koj tsuas paub qhov ntev ntawm ib sab thiab ntsuas ntawm lub kaum sab xis uas tsis yog lub kaum sab xis.
Kauj ruam
Txoj Kev 1 ntawm 3: Siv Pythagorean Theorem
Kauj Ruam 1. Kawm Pythagorean Theorem
Pythagorean theorem piav txog kev sib raug zoo ntawm ob sab ntawm daim duab peb sab xis. Cov theorem no hais tias rau txhua txoj cai daim duab peb sab nrog ob sab ntawm a thiab b, thiab hypotenuse raws c, a2 + b2 = c2.
Kauj Ruam 2. Nco ntsoov tias koj daim duab peb sab yog daim duab peb sab xis
Pythagorean theorem tsuas yog siv rau daim duab peb sab xis, thiab los ntawm kev txhais, tsuas yog daim duab peb sab txoj cai muaj qhov hypotenuse. Yog tias koj daim duab peb sab muaj ib lub kaum sab xis uas yog 90 degrees, nws yog daim duab peb sab xis thiab koj tuaj yeem txav mus.
Cov ces kaum sab xis feem ntau yog qhia hauv cov ntawv nyeem thiab kev xeem los ntawm ib lub xwmfab me me nyob rau ntawm lub ces kaum. Qhov cim tshwj xeeb no txhais tau tias "90 degrees"
Kauj Ruam 3. Muab cov kev hloov pauv a, b, thiab c rau ob sab ntawm koj daim duab peb sab
Qhov sib txawv "c" yeej ib txwm muab rau hypotenuse, lossis sab ntev tshaj plaws. Xaiv ib qho ntawm lwm sab los ua "a", thiab hu rau lwm sab "b" (nws tsis muaj teeb meem sab twg yog a lossis b; kev suav yuav zoo ib yam). Tom qab ntawd, ntsaws qhov ntev ntawm a thiab b rau hauv cov qauv, raws li qhov piv txwv hauv qab no:
Yog tias koj daim duab peb sab muaj ob sab ntawm qhov ntev 3 thiab 4, thiab koj tau muab cov ntawv sau rau ob sab kom a = 3 thiab b = 4, koj yuav sau koj cov zauv raws li: 32 + 42 = c2.
Kauj Ruam 4. Nrhiav lub xwmfab ntawm a thiab b
Txhawm rau nrhiav cov xwm txheej ntawm tus lej, koj tsuas yog muab tus lej los ntawm nws tus kheej, yog li ntawd a2 = x a aw. Nrhiav cov xwm txheej ntawm a thiab b, thiab ntsaws lawv rau hauv koj cov mis.
- Yog tias = 3, a2 = 3 x 3, lossis 9. Yog b = 4, b2 4x 4, or 16.
- Thaum koj ntsaws cov txiaj ntsig no rau hauv koj qhov kev ua zauv, koj qhov kev ua zauv yuav tsum zoo li no: 9 + 16 = c2.
Kauj Ruam 5. Ntxiv cov txiaj ntsig ntawm a2 thiab b2.
Txuas qhov sib npaug rau hauv koj qhov kev ua zauv, thiab qhov no yuav muab tus nqi c2. Tsuas muaj ib kauj ruam nkaus xwb, thiab koj yuav daws qhov hypotenuse!
Hauv peb qhov piv txwv, 9 + 16 = 25, yog li koj yuav sau 25 = c2.
Kauj Ruam 6. Nrhiav lub hauv paus c ntawm c2.
Siv cov hauv paus plaub fab ua haujlwm ntawm koj lub laij lej (lossis cim xeeb lossis koj cov lej sib npaug) kom pom cov hauv paus c ntawm c2. Cov lus teb yog qhov ntev ntawm koj qhov hypotenuse!
Hauv peb qhov piv txwv, c2 = 25. Lub hauv paus square ntawm 25 yog 5 (5 x5 = 25, yog li Hauv paus (25) = 5). Nws txhais tau tias, c = 5 hli, qhov ntev ntawm peb cov hypotenuse!
Txoj Kev 2 ntawm 3: Nrhiav Qhov Hypotenuse ntawm Ib Daim Duab Peb Sab Tshwj Xeeb
Kauj Ruam 1. Kawm paub txog daim duab peb sab nrog Pythagorean Triple
Qhov ntev ntawm Pythagorean peb npaug yog cov lej raws li Pythagorean Theorem. Cov duab peb sab tshwj xeeb no feem ntau tshwm hauv cov phau ntawv geometry thiab cov ntawv xeem zoo xws li UN. Yog tias koj nco tshwj xeeb tshaj yog thawj 2 Pythagorean peb zaug, koj tuaj yeem txuag sijhawm ntau ntawm cov kev ntsuas no vim tias koj yuav pom sai sai ntawm qhov hypotenuse ntawm ib qho ntawm cov duab peb sab no tsuas yog saib rau sab ntev!
- Thawj Pythagorean triple yog 3-4-5 (32 + 42 = 52, 9 + 16 = 25). Thaum koj pom daim duab peb sab xis nrog ob txhais ceg ntawm qhov ntev 3 thiab 4, koj yuav ntseeg tam sim ntawd tias nws qhov hypotenuse yog 5 yam tsis tas yuav suav ib qho lej.
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Pythagorean triple ratio tuav qhov tseeb txawm hais tias ob tog tau sib tw los ntawm lwm tus lej. Piv txwv li, daim duab peb sab xis nrog ceg ntev
Kauj ruam 6. ua da
Kauj ruam 8. yuav muaj hypotenuse
Kauj ruam 10. (62 + 82 = 102, 36 + 64 = 100). Tib yam mus rau 9-12-15, thiab txawm tias 1, 5-2-2, 5. Sim ua cov lej thiab pom ntawm koj tus kheej!
- Qhov thib ob Pythagorean peb zaug uas tshwm sim ntau zaus hauv kev xeem yog 5-12-13 (52 + 122 = 132, 25 + 144 = 169). Kuj xyuam xim rau cov zauv zoo li 10-24-26 thiab 2, 5-6-6, 5.
Kauj Ruam 2. Nco ntsoov qhov sib piv ntawm ob sab ntawm daim duab peb sab xis 45-45-90
Ib daim duab peb sab xis 45-45-90 muaj kaum ntawm 45, 45, thiab 90 degrees, thiab tseem hu ua isosceles txoj cai daim duab peb sab. Daim duab peb sab no tshwm sim ntau zaus hauv cov ntawv xeem, thiab yog daim duab peb sab yooj yim heev los daws. Qhov sib piv ntawm ob sab ntawm daim duab peb sab no yog 1: 1: Hauv paus (2), uas txhais tau hais tias qhov ntev ntawm ob txhais ceg zoo ib yam, thiab qhov ntev ntawm hypotenuse tsuas yog qhov ntev ntawm ob txhais ceg lub hauv paus ntawm ob.
- Txhawm rau suav qhov hypotenuse ntawm daim duab peb sab raws qhov ntev ntawm ib ntawm nws ob txhais ceg, tsuas yog muab qhov ntev ntawm txhais ceg los ntawm Sqrt (2).
- Paub txog cov kev sib piv no yog qhov muaj txiaj ntsig, tshwj xeeb tshaj yog thaum koj qhov kev kuaj mob lossis cov lus nug hauv tsev muab sab ntev raws li qhov hloov pauv hloov ntawm cov lej.
Kauj Ruam 3. Kawm sab piv ntawm 30-60-90 txoj cai daim duab peb sab
Cov voos plaub no muaj kev ntsuas kaum ntawm 30, 60, thiab 90 degrees, thiab tshwm sim thaum koj txiav ib daim duab peb sab sib npaug hauv ib nrab. Ob sab ntawm 30-60-90 txoj cai daim duab peb sab ib txwm muaj qhov sib piv 1: Hauv paus (3): 2, los yog x: Hauv paus (3) x: 2x. Yog tias koj tau muab qhov ntev ntawm ib txhais ceg ntawm txoj cai daim duab peb sab 30-60-90 thiab nug kom nrhiav tus hypotenuse, qhov teeb meem no yuav yooj yim heev ua:
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Yog tias koj tau muab qhov ntev ntawm ceg luv tshaj plaws (rov qab ntawm lub kaum sab xis 30-degree), tsuas yog muab qhov ntev ntawm txhais ceg los ntawm 2 txhawm rau nrhiav qhov ntev ntawm hypotenuse. Piv txwv, yog tias qhov ntev ntawm ceg luv tshaj plaws yog
Kauj ruam 4., koj paub tias qhov ntev ntawm hypotenuse yuav tsum yog
Kauj ruam 8..
- Yog tias koj tau muab qhov ntev ntawm txhais ceg ntev dua (piv rau lub kaum sab xis ntawm 60 degrees), sib npaug qhov ntev los ntawm 2/hauv paus (3) txhawm rau nrhiav qhov ntev ntawm hypotenuse. Piv txwv li, yog tias qhov ntev ntawm txhais ceg ntev dua
Kauj ruam 4., koj paub tias qhov ntev ntawm qhov tseeb hypotenuse yog 4, 62.
Txoj Kev 3 ntawm 3: Nrhiav Qhov Hypotenuse Siv Txoj Cai Sine
Kauj Ruam 1. Nkag siab lub ntsiab lus ntawm "Sine"
Cov ntsiab lus "sine", "cosine", thiab "tangent" hais txog ntau yam sib piv ntawm cov ces kaum thiab/lossis ob sab ntawm daim duab peb sab sab xis. tsi lub kaum sab xis yog txhais raws li qhov ntev ntawm sab ntxeev lub kaum sab xis faib los ntawm daim duab peb sab hypotenuse. Cov ntawv luv rau sine hauv kab zauv thiab cov laij lej yog kev txhaum.
Kauj Ruam 2. Kawm paub laij lej sine
Txawm tias cov lej laij lej yooj yim muaj lub sine muaj nuj nqi. Nrhiav lub pob uas hais tias kev txhaum. Txhawm rau nrhiav lub sine ntawm lub kaum ntse ntse, koj feem ntau nias tus yuam sij kev txhaum thiab tom qab ntawd nkag mus ntsuas lub kaum sab xis hauv degrees. Txawm li cas los xij, hauv qee lub laij lej, koj yuav tsum xub nkag mus ntsuas lub kaum ntse ntse thiab tom qab ntawd nyem lub pob kev txhaum. Koj yuav tau sim nrog koj lub laij lej lossis tshawb xyuas phau ntawv kom paub seb yuav siv txoj hauv kev twg.
- Txhawm rau nrhiav sine ntawm lub kaum sab xis 80 degree, koj yuav tsum nkag mus sib 80 ua raws li qhov kos npe sib npaug lossis Sau, lossis 80i tug. (Cov lus teb yog -0, 9939.)
- Koj kuj tseem tuaj yeem ntaus "sine calculator" rau hauv kev tshawb nrhiav lub vev xaib, thiab nrhiav qee qhov yooj yim-rau-siv cov laij lej, uas yuav siv qhov kev kwv yees tawm ntawm txoj kev.
Kauj Ruam 3. Kawm Txoj Cai Sine
Txoj cai ntawm sines yog cov cuab yeej muaj txiaj ntsig zoo rau kev daws teeb meem peb tog. Tshwj xeeb, txoj cai no tuaj yeem pab koj nrhiav qhov hypotenuse ntawm daim duab peb sab yog tias koj paub qhov ntev ntawm ib sab, thiab ntsuas ntawm ib lub kaum sab xis uas tsis yog lub kaum sab xis. Rau txhua daim duab peb sab nrog sab a, b, thiab c, thiab cov ces kaum A, B, thiab C, Txoj Cai Sine hais tias ib / sin A. = b / sin IB = c / sib C.
Txoj cai ntawm sines tuaj yeem siv tau los daws txhua daim duab peb sab, tab sis tsuas yog cov voos plaub sab xis muaj qhov hypotenuse
Kauj Ruam 4. Muab cov kev hloov pauv a, b, thiab c rau ntawm ob sab ntawm koj daim duab peb sab
Qhov hypotenuse (sab ntev tshaj) yuav tsum yog "c". Txhawm rau kom yooj yim, daim ntawv lo "a" rau sab ntawm qhov paub ntev, thiab daim ntawv lo "b" rau lwm sab. Lub kaum sab xis sab xis tiv thaiv hypotenuse yog "C". Lub kaum sab xis rov qab "a" yog lub kaum sab xis "A", thiab lub kaum sab xis rov qab "b" yog "B".
Kauj Ruam 5. Xam qhov ntsuas ntawm lub kaum peb
Txij li nws yog lub kaum sab xis, peb twb paub lawm C = 90 degrees, thiab koj kuj paub qhov ntsuas A los yog B. Txij li kev ntsuas ntsuas sab hauv ntawm daim duab peb sab ib txwm sib npaug 180 degrees, koj tuaj yeem suav tau yooj yim ntsuas qhov ntsuas ntawm txhua lub kaum peb siv cov qauv: 180 - (90 + A) = B. Koj tseem tuaj yeem thim qhov sib npaug rau 180 - (90 + B) = A.
Piv txwv li, yog koj paub qhov ntawd A = 40 qib, B = 180 - (90 + 40). Yooj yim rau qhov no IB = 180 - 130, thiab koj tuaj yeem txiav txim siab sai sai B = 50 qib.
Kauj Ruam 6. Txheeb xyuas koj daim duab peb sab
Hauv qib no, koj twb paub qhov ntsuas ntawm peb lub kaum sab xis, thiab qhov ntev ntawm sab a. Tam sim no nws yog lub sijhawm los ntsaws cov ntaub ntawv no rau hauv Txoj Cai ntawm Sine cov kab zauv los txiav txim qhov ntev ntawm ob sab.
Txhawm rau txuas ntxiv peb qhov piv txwv, cia peb hais qhov ntev ntawm a = 10. Lub kaum sab xis C = 90 degrees, lub kaum sab xis A = 40 degrees, thiab kaum B = 50 degrees
Kauj Ruam 7. Siv Txoj Cai Sine rau koj daim duab peb sab
Peb tsuas yog yuav tsum ntsaws peb tus lej hauv thiab daws cov kab zauv hauv qab no kom pom qhov ntev ntawm hypotenuse c: sab ntev a / sin A = sab ntev c / sin C. Qhov sib npaug no yuav saib me ntsis txaus ntshai, tab sis sine ntawm 90 degrees ib txwm zoo ib yam, thiab ib txwm sib npaug 1! Yog li, peb qhov sib npaug tuaj yeem yooj yim rau: a / sin A = c / 1, los yog xwb a / sin A = c.
Kauj Ruam 8. Faib qhov ntev ntawm sab a nrog lub sine ntawm lub kaum sab xis A txhawm rau nrhiav qhov ntev ntawm hypotenuse!
Koj tuaj yeem pom nws hauv ob qib sib cais, ua ntej los ntawm kev suav qhov txhaum A thiab sau qhov txiaj ntsig, tom qab ntawd faib los ntawm a. Lossis koj tuaj yeem nkag tau txhua yam rau hauv lub laij lej tib lub sijhawm. Yog tias koj tab tom siv lub laij lej, nco ntsoov muab cov lus sib hais tom qab faib npe. Piv txwv li, nkag mus 10 / (txhaum 40) los yog 10 / (40 puas), nyob ntawm koj lub laij lej.
