Nheyo dzeTrigonometry dzeVatangi: Kunzwisisa Mativi neMakona eTriangle
Trigonometry ibazi remasvomhu rinoongorora hukama huripo pakati pemativi nemakona ematriangles. Izwi rekuti "trigonometry" rinobva pamashoko echiGiriki ekuti "trigonon," zvichireva kuti "triangle," uye "metron," zvichireva kuti "measure." Muchinyorwa chino, tichakurukura nezvekutanga kwetrigonometry, izvo zvakakosha kuti vanotanga vanzwisise pfungwa dzakanyanya kukura mumasvomhu nefizikisi.
1. Chii chinonzi Trigonometry?
Trigonometry inotarisa pamatriangles, kunyanya matriangles ekurudyi, uko kona imwe iri madhigirii makumi mapfumbamwe. Hukama huripo pakati pemativi etriangle nemaangles ayo hunomiririrwa nemabasa ekutanga etrigonometric: sine (sin), cosine (cos), uye tangent (tan).
Basa rimwe nerimwe rinotarisa pahurefu hwemativi ari mutriangle yekurudyi.
– Sine (sin) yekona ireshiyo iri pakati pehurefu hwedivi rakapesana nehurefu hwehypotenuse.
– Cosine (cos) yekona ihuremu huri pakati pehurefu hwedivi riri pedyo nehurefu hwehypotenuse.
– Tangent (tan) yekona ihuremu huri pakati pehurefu hwedivi rakapesana nehurefu hwedivi riri pedyo.
Kuti tinzwisise zviri nani, ngatifungei nezvetriangle yekurudyi ine kona θ:
``
|\
| \
o | \ h
| \
|—-|
a
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Information:
– o kureba kwedivi rakapesana kana rakapesana rekona θ.
– a ndiko kureba kwerutivi kana divi riri pedyo rekona θ.
– h ndiyo kureba kwe hypotenuse ye angle θ.
Saka, tinogona kunyora mabasa ekutanga e trigonometric seizvi:
– chivi(θ) = o / h
– cos(θ) = a / h
– tan(θ) = o / a
2. Sei Trigonometry Yakakosha?
Trigonometry inoshandiswa zvakanyanya muzvikamu zvakasiyana-siyana zvedzidzo uye zvehupenyu hwezuva nezuva. Senzira yekutanga, trigonometry inowanzoshandiswa mu:
- Fizikisi: Kuyerwa kwemafungu, kuongorora simba, uye kufamba kwenguva nenguva.
– Kuongorora nyeredzi: Kuyera daro riri pakati penyeredzi uye kuverenga kutenderera kwemapuraneti.
– Mainjiniya: Kugadzira zvivakwa, mabhiriji, nezvimwe zvivakwa.
- Kufamba: Kuziva nzvimbo uye gwara rechikepe kana ndege.
3. Kuzivikanwa kweTrigonometric
Kuwedzera kumabasa ekutanga etrigonometric, kune marudzi akasiyana-siyana etrigonometric ayo akakosha mukuita kuti kuverenga kuve nyore:
- Kuzivikanwa kwePythagoras:
sin²(θ) + cos²(θ) = 1
- Kuzivikanwa kweSum uye Musiyano weAngles:
chivi(A + B) = chivi(A)cos(B) + cos(A)chivi(B)
cos(A + B) = cos(A)cos(B) - chivi(A)chivi(B)
- Kuzivikanwa Kuviri:
chivi(2θ) = 2chivi(θ)cos(θ)
cos(2θ) = cos²(θ) - sin²(θ)
Kuzivikanwa uku kunopa nzira dzakakosha dzekushandura nekurerutsa matauriro emasvomhu.
4. Nhanganyaya yeZvikamu zveAngle
Kunyange hazvo madhigirii ari iwo anonyanya kushandiswa pakuyera maangles, trigonometry inoshandisawo ma radians. Denderedzwa rakazara i360 degrees, kana kuti 2π radians. Saka, radian imwe yakaenzana ne180/π degrees (inenge 57,2958 degrees).
Kuchinja pakati pemadhigirii nemaradians kunogona kuitwa seizvi:
– Shandura kubva pamadhigirii kuenda kumaradians: maradians = madhigirii × (π / 180)
– Shandura kubva kuradians kuenda kudigirii: madhigirii = radians × (180 / π)
5. Magirafu ebasa reTrigonometric
Kugadzira girafu yema "trigonometric functions" kunobatsira kuona kuti anochinja sei kukosha zvichienderana nekona. Heino chidimbu chegirafu yakadaro:
- Girafu Yekuyera:
Girafu ye sine wave iwave inodzokororwa ine nguva ye 2π. Nheyo yepamusoro (kukosha kwepamusoro) iri pa 1 uye mupata (kukosha kushoma) uri pa -1.
- Girafu yeCosine:
Girafu recosine rakafanana negirafu resine asi rinotanga ne1 pakona 0. Kufanana nesine, girafu iyi inodzokororawo 2π yega yega.
- Girafu yeTangent:
Girafu ine tangent inoratidza shanduko inoshamisa, ichidzokorora π yega yega. Kukosha kwayo kunosvika pakusaguma painoswedera pamadhigirii makumi mapfumbamwe ne -90, zvichiita kuti pave neasymptote yakatwasuka.
6. Kushandiswa kweTrigonometry: Kugadzirisa Matambudziko
Ngatitarisei muenzaniso wekushandiswa kwetrigonometry muhupenyu chaihwo:
Muenzaniso 1: Kuverenga Kureba kweChivako
Ngatitii munhu akamira mamita makumi mashanu kubva pasi pechivako uye anoyera kona yekukwirira kusvika pamusoro pechivako semadhigirii makumi matatu. Chivako chacho chakareba sei?
Kubva pano, tinoshandisa basa re tangent:
tan(θ) = mhiri / pedyo
tan(30°) = awa / 50
h = 50 tan(30°)
h ≈ 50 0.577 = 28.85 mamita
Saka, kukwirira kwechivako ichi kuri mamita 28.85.
Muenzaniso 2: Kushandisa Trigonometric Identities
Sarudza kukosha kwechivi(2θ) kana chivi(θ) = 3/5 uye cos(θ) = 4/5.
Tinoshandisa hunhu:
chivi(2θ) = 2chivi(θ)cos(θ)
chivi(2θ) = 2 (3/5) (4/5)
chivi(2θ) = 24/25 = 0.96
Saka, chivi(2θ) = 0.96.
Penutup
Trigonometry chishandiso chemasvomhu chine simba guru uye chinochinjika. Kubva pakunzwisisa kufamba kwemapuraneti kusvika pakugadzira zvivakwa, trigonometry inopa mutauro nenzira yekuongorora nekunzwisisa nyika yakatipoteredza. Nekuziva izvi zvakakosha zvetrigonometry, vanotanga vanogona kuvhura mukana wekuwana ruzivo rwakakura uye mashandisirwo anoshanda muzvidzidzo zvakasiyana-siyana.
