Mutemo weCosines: Kushandiswa, Mienzaniso uye Kudzidzira

Mutemo weCosines, unozivikanwawo seMutemo weCosines, chishandiso chakakosha chesvomhu mutrigonometry chinobvumira kureba kwerimwe divi rekona yekona risiri yekurudyi kutariswa pachishandiswa kukosha kwemamwe mativi maviri uye kona iri pakati pawo. Mutemo uyu unoshandiswa zvakanyanya mumapazi akasiyana einjiniya nefizikisi, uchipa mhinduro chaiyo uye inoshanda yekugadzirisa matambudziko akaomarara ejometri. Muchikamu chino, tichaongorora mashandisirwo ayo zvakadzama. mienzaniso nemabasa ekudzidzira kushandiswa kunoshanda kweMutemo weCosines, kupa vaverengi kunzwisisa kwakasimba kweichi chishandiso chesvomhu uye kubatsira kwacho mumamiriro akasiyana ehunyanzvi.

1. Nhanganyaya kuMutemo weCosines: Kushandiswa mumatambudziko ejometri

Mutemo weCosines chishandiso chakakosha mujometry kugadzirisa matambudziko zvinoenderana nematatu. Mutemo uyu unoti sikweya yerimwe divi regonyo yakaenzana nehuwandu hwemakona emamwe mativi maviri kubvisa kaviri chigadzirwa che los dos lados necosine yekona yakatarisana nedivi iro. Nekunzwisisa nekushandisa mutemo uyu nemazvo, tinogona kugadzirisa matambudziko akasiyana-siyana ejometri.

Kuisa Mutemo weCosine kumatambudziko ejometri, danho rekutanga ndere kuona gonyo riri mubvunzo uye kuisa mativi aro nemakona. Zvadaro, tinofanira kuona kuti ndeupi ruzivo rwatinopiwa uye ruzivo rwatinokumbirwa kuwana. Kubva ipapo, tinogona kushandisa Mutemo weCosines formula kugadzirisa dambudziko. Zvakakosha kuyeuka kushandura makona kuita radians kana zvichidikanwa usati waisa fomula.

Zano rinobatsira kana uchishandisa Mutemo weCosine kuparadzanisa zvisingazivikanwe kana divi ratinoda kuwana kubva mufomula tisati tatsiva kukosha. Izvi zvichaita kuti zvive nyore kugadzirisa equation uye kudzivirira kukanganisa mukuverenga. Zvinogona kubatsirawo kushandisa mabasa etrigonometric uye zvimiro zvemakona matatu (sekuwanda kwemakona emukati) kurerutsa dambudziko nekutsvaga hukama pakati pemativi nemakona. Kushandisa trigonometric calculation maturusi kunogonawo kubatsira zvakanyanya mukuona zvawanikwa.

2. Tsanangudzo yemasvomhu yeMutemo weCosines uye maitiro ayo

Mutemo weCosines chishandiso chesvomhu chinobatsira chekugadzirisa matrietu asiri ekurudyi. Mutemo uyu unogadza hukama pakati pehurefu hwemativi egonyo nemakona akatarisana nawo. Mutemo weCosines formula inogona kushandiswa kutsvaga kureba kwemativi egonyo nemakona anoenderana.

Iyo formula yeMutemo weCosines inoratidzwa seinotevera:
c^2 = a^2 + b^2 – 2ab * cos(C)

Apo "a" na "b" ndiwo hurefu hwemativi maviri anozivikanwa, "C" ikona yakatarisana nedivi risingazivikanwi "c," uye "cos" rinoreva basa re cosine. Iyi fomula inokutendera kuti uwane kukosha kwemativi nemakona egonyo kana uchinge waziva kureba kwemativi maviri neakatarisana.

Kushandisa Mutemo weCosines, matanho anotevera anofanira kuteverwa:
1. Ziva kukosha kwemativi anozivikanwa uye kona yakatarisana nedivi risingazivikanwe.
2. Kutsiva kukosha muMutemo weCosines formula.
3. Gadzirisa equation kuti uwane kukosha kwedivi risingazivikanwe kana kona inoenderana.
4. Dzokorora nzira kana zvichidiwa kune mamwe mativi kana makona.

Zvakakosha kuziva kuti makona anofanirwa kuyerwa mumaradians kuti ushandise Mutemo weCosines formula, saka ungangoda kuashandura kana aratidzwa mumadhigirii. Zvakare, paunenge uchishandisa fomula, teerera kune zvakanaka uye zvisina kunaka zviratidzo muyekupedzisira equation kuti uve nechokwadi chekuti unowana kukosha kwayo. Kushandisa Mutemo weCosines kunogona kubatsira zvakanyanya mukugadzirisa matambudziko anosanganisira asiri ekurudyi matatu matatu uye kuona maitiro avo.

3. Kushandiswa kweMutemo weCosines muscalene uye obtuse triangles

Mutemo weCosines idzidziso yakakosha inoshandiswa mujometry kugadzirisa matambudziko anosanganisira scalene uye obtuse katatu. Mutemo uyu unogadza hukama pakati pemativi nemakona egonyonhatu, zvichiita kuti tikwanise kuziva maitiro asingazivikanwi. Pazasi pane matanho anodiwa kuti ushandise nemazvo Mutemo weCosine kune aya marudzi ematatu.

Danho 1: Ziva scalene kana obtuse triangle. Ita shuwa kuti katatu haina kuenzana kana kurudyi, sezvo paine mafomula chaiwo enyaya idzodzo.

Danho rechipiri: Ziva maitiro aripo. Kuti ushandise Mutemo weCosines, unofanirwa kuziva zvinokwana zvitatu zvezvinhu zvitanhatu zvegonyo: mativi nemakona akatarisana nemativi iwayo.

4. Mienzaniso yezvinetso zvinogadziriswa uchishandisa Mutemo weCosines

5. Kuverenga kwemativi asingazivikanwe nemakona neMutemo weCosines

Mutemo we cosines chishandiso chakakosha chekuverenga mativi asingazivikanwe nemakona mune matatu. Mutemo uyu unogadza hukama pakati pemativi egonyo nemakona akatarisana nawo. Kuti ushandise mutemo we cosines, unoda ruzivo pamusoro pezvinhu zvinenge zvitatu zvetatu: mativi maviri uye kona pakati pavo.

Danho rekutanga mukushandisa mutemo wecosine nderokuziva izvo zvinozivikanwa uye zvisingazivikanwe zvinhu zvegonyo. Zvadaro, fomura yakakodzera ichashandiswa kutsvaga kukosha kwedivi risingazivikanwe kana kona. Iyo general formula yemutemo we cosines ndeiyi:

c^2 = a^2 + b^2 – 2ab * cos(C)

Apo "c" ndiro divi risingazivikanwi, "a" na "b" ndiwo mativi anozivikanwa, uye "C" ikona yakatarisana ne "c." Kuti ushandise iyi fomula, unofanirwa kugadzirisa kune iro risingazivikanwe divi kana kona uye kuita maverengero anodiwa. Zvinokurudzirwa kushandisa karukureta yesainzi kuti uwane mibairo chaiyo.

6. Nyaya dzinokosha dzeMutemo weCosines: kurudyi uye isosceles triangles

6. Nyaya dzinokosha dzeMutemo weCosines: kurudyi uye isosceles triangles

Mutemo weCosines, unobatanidza mativi egonyo nemakona akapesana, chishandiso chine simba mukugadzirisa matambudziko ejometri. Nekudaro, pane zviitiko zvakakosha apo mutemo uyu unogona kurerutswa kuti ufambise maverengero. Muchinyorwa chino, isu tichatarisa pane maviri chaiwo kesi: iyo yekona yekona uye isosceles triangle.

Triángulo rectángulo

Gonyonhatu yekurudyi ndiyo ine kona imwe yemukati Madhigirii 90Mumamiriro ezvinhu aya, Mutemo weCosines unoderedza kune inozivikanwa Pythagorean formula. Kuti tiwane kureba kwerimwe mativi, isu tinongoshandisa formula:

a² = b² + c²

Apo "a" inonzi hypotenuse (rutivi rwakarebesa rwegonyonhatu) uye "b" na "c" ndiwo makumbo (mamwe mativi maviri). Iyi fomula inobatsira zvikuru mumatambudziko anosanganisira kureba kwemativi ekona yekona, sezvo ichirerutsa zvikuru maverengero anodiwa.

Triángulo isósceles

Isosceles triangle ndiyo ine mativi maviri akaenzana. Muchiitiko ichi, Mutemo weCosine unowedzera kurerutswa. Kana tichiziva hurefu hwemativi maviri akaenzana (a) uye kona yakaumbwa navo (θ), tinogona kuwana hurefu hwedivi rasara (b) tichishandisa fomula inotevera:

b = 2a * cos(θ / 2)

Iyi fomula inobatsira kana isu tichida kuona kureba kwerimwe mativi mune isosceles triangle pasina kushandisa general Law yeCosines formula. Inorerutsa maverengero uye inotitendera kuti tiwane mibairo chaiyo nemazvo.

7. Kugadzirisa zviitwa zvinoshanda uchishandisa Mutemo weCosines

Kugadzirisa zviitwa zvinoshanda uchishandisa Mutemo weCosines, zvakakosha kutevedzera anotevera matanho:

1. Ongorora dambudziko: Verenga chirevo chechiitwa zvakanyatsonaka kuti unzwisise zviri kubvunzwa uye kujekesa nezve data rakapihwa.
2. Ziva zvinhu: Ziva mativi nemakona egonyonhatu iri kutaurwa wogovapa mavara kana zviratidzo zvinofambirana.
3. Shandisa fomula: Mutemo weCosines unoti sikweya yerimwe divi regonyo yakaenzana nehuwandu hwemakwere emamwe mativi maviri, kubvisa kaviri chigadzirwa chehukuru hwemativi iwayo akapetwa necosine yekona yakatarisana. Tichishandisa iyi fomula, tinogona kugadzirisa chiitiko nhanho nhanho.

Zvakakosha kurangarira kushandura chero kona inotaridzwa mumadhigirii kuita radians usati waita masvomhu. Scientific calculator kana maturusi epamhepo anogona zvakare kushandiswa kufambisa anodiwa trigonometric kuverenga.

Muenzaniso unopiwa pazasi kuratidza maitiro:

1. Ngatitii tine gonyotatu ABC, apo divi rinoyera mayunitsi masere, divi b rinoyera mayunitsi gumi, uye kona C yakatarisana nedivi c inoyera 8°.
2. Tinoona zvinhu: a = 8, b = 10, uye angle C = 45 °.
3. Isu tinoshandisa fomula: c² = a² + b² - 2ab * cos(C)
4. Isu tinotsiva maitiro anozivikanwa: c² = 8² + 10² - 2(8)(10) * cos(45°)
5. Tinoverenga cos(45°) = √2 / 2 ≈ 0.707
6. Tinoenderera mberi neformula: c² ≈ 64 + 100 – 2(8) (10) * 0.707
7. Isu tinoita mabasa: c² ≈ 64 + 100 - 113 ≈ 51
8. Chekupedzisira, tinoona kukosha kwe c nekutora square root ye mativi ese ari maviri: c ≈ √51 ≈ 7.14 zvikamu

Nekutevera nhanho idzi uye nekuchengetedza yakaomarara nzira, zvinokwanisika kugadzirisa zvinobudirira maekisesaizi uchishandisa Mutemo weCosines.

8. Kushandiswa kweMutemo weCosines mukufamba kwegungwa uye nyeredzi

Mutemo weCosines chishandiso chakakosha chinoshandiswa mukufamba kwemugungwa uye nyeredzi kuverenga madaro nemakona mune asiri ekurudyi matatu matatu.

Mukufamba kwemugungwa, Mutemo weCosines unoshandiswa kuona chinhambwe uye gwara pakati pemapoinzi maviri pane nautical chart. Kuziva makona uye kureba kweparutivi kwegonyonhatu kwakaumbwa nenzvimbo dzekutanga nedzinogumira zvinobvumira vafambisi vezvikepe kuronga nzira dzakasimba uye kudzivirira zvipingamupinyi. Kuti ushandise mutemo uyu, zvinodikanwa kuve neruzivo rwakakwana nezve kurongeka kwenzvimbo yemapoinzi uye kushandisa mafomula chaiwo anosanganisira kushandiswa kwemakosini.

Mukuongorora nyeredzi, Mutemo weCosines unoshandiswa kuverenga chinhambwe pakati pezvitunha zviviri zvekudenga, semapuraneti kana nyeredzi. Kuziva madaro aya kwakakosha pakuona nzvimbo yavo muchadenga uye kufanotaura mafambiro avo. Nyanzvi dzemuchadenga dzinoshandisa mafomula anoenderana neMutemo weCosines kuverenga ukoshi uhwu, kubatanidza kuyerwa kwemakona uye nhambwe inowanikwa kuburikidza neteresikopu ine simba. Kuti uwane mhedzisiro chaiyo, zvakakosha kushandisa data rakavimbika uye kushandisa maverengero nemazvo uye nenzira.

Muchidimbu, Mutemo weCosines chishandiso chakakosha mukufamba kwemugungwa uye nyeredzi pakuverenga madaro nemakona mune asiri ekurudyi matatu matatu. Kushandiswa kwaro kunoda ruzivo rwemafomu chaiwo uye kushandiswa kwedata rakarurama. Vese vafambisi vezvikepe nevanoongorora nyeredzi vanoshandisa mutemo uyu kuita zviverengero zvakakosha muzvidzidzo zvavo uye kuwana mhinduro dzakavimbika.

9. Kushandiswa kweMutemo weCosine kuona madaro uye kureba

Mutemo weCosines inzira inobatsira kwazvo yekuona madaro uye kureba mumatambudziko ejometry. Mutemo uyu unoshandiswa panozivikanwa kureba kwemativi maviri nekona pakati pawo, kana kuti panozivikanwa mativi ose matatu egonyo. Izvi zvinotevera zvinotsanangura nhanho-ne-nhanho nzira yekushandisa mutemo uyu kugadzirisa dambudziko.

1. Chekutanga, ziva iyo data yauri kupihwa uye iyo data yauri kukumbirwa. Ita shuwa kuti wanyora zvese muchikamu chimwe chekuyera. Kana makona ari mumadhigirii, shandura iwo kuita radians.

2. Shandisa Mutemo weCosine kuti uwane huwandu husingazivikanwe. Iyo general formula yeMutemo weCosines ndeiyi: c² = a² + b² - 2ab*cos(C). Apo 'c' pane hurefu husingazivikanwe, 'a' na'b' ndiwo hurefu hunozivikanwa, uye 'C' ndiyo kona iri pakati pemativi anozivikanwa. Kana iwe uchiziva mativi matatu ese matatu, unogona kushandisa fomula kuti uwane kona isingazivikanwe: cos(C) = (a² + b² - c²) / (2ab).

10. Kudzidzira kwekushandiswa kweMutemo weCosine mumatambudziko chaiwo ehupenyu

Muchikamu chino, tichakuratidza mamwe . Mutemo weCosines chishandiso chesvomhu chinotibvumira kuverenga kureba kwedivi regonyo kana tichiziva mamwe mativi maviri uye kona yakatarisana nedivi iro.

1. Basa rekutanga: Ngatitii tinoda kuona nhambwe iri pakati pemapoinzi maviri pamepu. Sezvo tichingokwanisa kuwana mepu ine mativi maviri, hatigone kuyera chinhambwe mumutsara wakatwasuka. Pane kudaro, tinofanira kushandisa Mutemo weCosines. Kugadzirisa dambudziko iri, tinofanira kutanga taziva mativi maviri anozivikanwa uye kona yakatarisana nedivi risingazivikanwi. Zvadaro, isu tinoshandisa Mutemo weCosines fomula kuti tiwane kureba kwedivi risingazivikanwe.

2. Exercise 2: Fungidzira uri kuvaka rampu yekupinda papuratifomu yakakwirira. Iwe unoziva kureba kwerampu uye kureba kwainofanira kukwira, asi iwe unofanirwa kuona kona iyo rampu inofanirwa kuburitsa yakaringana kutenuka. Kugadzirisa dambudziko iri, tinogona kushandisa Mutemo weCosines. Kuziva hurefu hwerampu uye kureba kwayo, tinogona kuwana kona yakatarisana nerampu tichishandisa Mutemo weCosines formula. Izvi zvinotitendera kuti tigadzire rampu ine mutsetse chaiwo.

3. Exercise 3: Parwendo rwechikepe, unoda kuverenga nhambwe iri pakati pemapoinzi maviri panyanza. Iwe unokwanisa kuwana GPS iyo inokupa iwe latitude uye longitude yemapoinzi ese ari maviri. Nekudaro, mapoinzi haana kuwanikwa mumutsara wakatwasuka, uye haugone kuyera chinhambwe chakananga pamepu. Kugadzirisa dambudziko iri, unogona kushandisa Mutemo weCosines. Uchishandisa Mutemo weCosines formula, unogona kuverenga chinhambwe chiri pakati pemapoinzi maviri uchishandisa iwo latitude uye longitude makongiresi.

Kushandisa Mutemo weCosines kumatambudziko ehupenyu chaihwo kunotipa chishandiso chine simba chesvomhu chekugadzirisa mamiriro anoshanda. Nekutevera maekisesaizi aya uye nzira yakatsanangurwa, unozokwanisa kuverenga kureba kusingazivikanwe, kuona makona, uye kufungidzira madaro mumamiriro akasiyana. Ongorora izvi zviitwa uye uwane hunyanzvi hutsva mu geometry uye trigonometry!

11. Dambudziko rematambudziko epamusoro anoda kushandiswa kweMutemo weCosines

Kugadzirisa matambudziko epamberi anoda kushandiswa kweMutemo weCosines, zvakakosha kutevedzera akatevedzana matanho kuti uwane mhinduro kwayo. Heino nhanho-ne-nhanho gwara rekukubatsira kukurira dambudziko iri:

Danho rekutanga: Nzwisisa dambudziko. Verenga chirevo zvakanyatsonaka uye uve nechokwadi chekuti unonzwisisa zviri kubvunzwa uye ruzivo rwupi rwuri kupihwa. Ziva makona nemativi aunoziva uye ndeapi aunoda kuwana.

Danho rekutanga: Shandisa Mutemo weCosines. Mutemo uyu unoti sikweya yerimwe divi regonyo yakaenzana nehuwandu hwemakona emamwe mativi maviri kubvisa kaviri chigadzirwa chemativi iwayo uye cosine yekona yakatarisana. Shandisa iyi fomula kumisa equation yaunokwanisa kugadzirisa kuti uwane iyo isingazivikanwe kukosha.

Danho rekutanga: Gadzirisa equation uchishandisa pfungwa dzemakona netrigonometry. Ungangoda kuisa matrigonometric identity kana kushandisa inverse trigonometric function kuti uwane kukosha kwekona kana divi risingazivikanwe. Kana zvichidikanwa, shandisa karukureta yesainzi kuita masvomhu.

12. Zvakanakira uye zvisingakwanisi kushandisa Mutemo weCosines mu trigonometric calculations

13. Hukama pakati peMutemo weCosine uye mamwe trigonometric geometry mafomula

1. Mutemo weCosines: Mutemo weCosines iformula yakakosha mutrigonometric geometry inotitendera kuverenga divi kana kona yechero katatu. Mutemo uyu unoti sikweya yerimwe divi regonyo yakaenzana nehuwandu hwemakona emamwe mativi maviri, kubvisa kaviri chigadzirwa chemativi aya uye cosine yekona yakatarisana nedivi iro.
2. Hukama nePythagorean Theorem: Mutemo weCosines ndeye generalization yePythagorean Theorem, sezvo kana katatu iine rectangular uye imwe yemukati makona inoyera 90 madhigirii, iyo cosine yekona iyo ichave yakaenzana ne zero uye fomula yeMutemo weCosines inodzikira kune formula yePythagorean Theorem.
3. Ejemplos de aplicación: Mutemo weCosines unobatsira zvikuru mumamiriro ezvinhu apo hutsika hwemativi maviri etatu uye kona pakati pawo zvinozivikanwa, kana apo hutsika hwemativi ose matatu hunozivikanwa uye imwe yemakona inodiwa. Semuenzaniso, kana tiine gonyonhatu ine mativi ehurefu 5, 7, uye 9 mauniti, tinogona kushandisa Mutemo weCosine kuverenga kona yakatarisana nedivi rehurefu 7. Kuti tiite izvi, tinoshandisa Mutemo weCosines fomula, kutsiva inozivikanwa tsika, uye kugadzirisa inoguma equation.

14. Mhedziso pamusoro pekukosha uye kubatsira kweMutemo weCosines munzvimbo dzakasiyana dzekudzidza nekuita.

Muchidimbu, Mutemo weCosines chishandiso chakakosha munzvimbo dzakasiyana siyana dzekufunda nekuita, senge trigonometry, fizikisi, engineering, uye cartography. Mutemo uyu unotibvumira kugadzirisa matambudziko anosanganisira makona matatu asiri ekurudyi, uchitipa nzira yakarurama yekuverenga mativi kana makona asingazivikanwe. Iyo general formula, c^2 = a^2 + b^2 – 2ab * cos(C), inotipa hwaro hwakasimba hwekugadzirisa akasiyana masvomhu uye geometric mamiriro.

Kubatsira kweMutemo weCosines kuri mukukwanisa kwayo kugadzirisa matriangles kubva kune rumwe ruzivo ruzivo, iro rinonyanya kubatsira mumamiriro ezvinhu apo asiri ese mativi kana makona anozivikanwa. Uyezve, nekuda kwemutemo uyu, tinogona kuona kuvapo kwekatatu uye chimiro chayo, kunyange mumamiriro ezvinhu apo Pythagorean Theorem haigoni kushandiswa. Naizvozvo, mashandisirwo ayo anosvika kumatambudziko ekufamba, geolocation, dhizaini dhizaini, kuverenga kwekumanikidza, uye dzimwe nzvimbo dzakawanda.

Mukupedzisa, Mutemo weCosines unoratidza kuve chishandiso chakakosha uye chinogoneka munzvimbo dzakasiyana dzekudzidza nekuita. Chimiro chayo chinotibvumira kugadzirisa matatu matatu asina kururama nemazvo uye nemazvo, achitipa kunzwisisa kwakadzama kwehukama pakati pemativi avo nemakona. Kushandisa Mutemo weCosines kunotibatsira kuita masvomhu uye kuongorora munzvimbo dzakasiyana sekugadzira mepu, kuona madaro nemakona mukuongorora nyeredzi, nekugadzirisa matambudziko ejometri muinjiniya. Kuziva mutemo uyu kwakakosha kuti ubudirire kugadzirisa mamiriro akasiyana emasvomhu uye ejometri mubasa redu redzidzo uye rinoshanda.

Mukupedzisa, Mutemo weCosines chinhu chakakosha chesvomhu mu trigonometry chinobvumira asiri-kurudyi matatu matatu kuti agadziriswe nemazvo uye nemazvo. Kushandisa kwayo kwakakosha munzvimbo dzakasiyana siyana, senge engineering, fizikisi, uye kufamba.

Uchishandisa Mutemo weCosines formula, zvinokwanisika kuverenga kureba kwechikamu chisingazivikanwe chetatu, pamwe nekuona makona ayo emukati. Izvi zvinowanikwa kuburikidza nekushandisa kuyerwa kwemativi anozivikanwa uye makona, ayo anobatsira kugadzirisa matambudziko akaoma mugeometry ndege.

Kuburikidza nenhevedzano yemienzaniso uye maekisesaizi anoshanda, takaratidza mashandisiro eMutemo weCosines kugadzirisa matambudziko chaiwo ehupenyu. Kubva pakuona nhambwe pakati pemapoinzi maviri mundege kusvika pakuverenga gwara chechinhu Mukufamba, chishandiso chesvomhu chine simba ichi chinopa mhinduro chaidzo uye dzakavimbika.

Kunzwisisa Mutemo weCosines kwakakosha kune chero mudzidzi kana nyanzvi inoshuvira kupinda munyika inonakidza ye trigonometry. Kubata iyi fomula inokupa iwe kugona kugadzirisa matambudziko akaomarara ejometri, kukwidziridza mapurojekiti einjiniya, uye kuita mamwe chaiwo maverengero muzvidzidzo zvakasiyana.

Muchidimbu, Mutemo weCosines unomiririra mbiru yakakosha yetrigonometry uye unopa nyanzvi kugona kugadzirisa matambudziko ejometri zvinesimba. Kushandiswa kwaro uye kudzidzira kusingachinji kunosimbisa hunyanzvi hwemasvomhu uye kunopa kunzwisisa kwakadzama kwenyika yakatitenderedza. Pasina kupokana, mutemo uyu chishandiso chine simba chekusimudzira kwesainzi uye tekinoroji munharaunda yedu nhasi.