Ngā Kaupapa Taketake o te Pānga-toru mō ngā Tīmatanga: Te Mārama ki ngā Taha me ngā Koki o te Tapatoru
Ko te ine whārite he peka o te pāngarau e ako ana i ngā whanaungatanga i waenga i ngā taha me ngā koki o ngā tapatoru. Ko te kupu "ine whārite" i ahu mai i ngā kupu Kariki "trigonon," ko tōna tikanga he "tapatoru," me te "metron," ko tōna tikanga he "ine." I roto i tēnei tuhinga, ka hipokina e mātou ngā kaupapa matua o te ine whārite, he mea nui mō ngā tīmatanga ki te mārama ki ngā ariā matatau ake o te pāngarau me te ahupūngao.
1. He aha te Pānga-toru?
Ko te aro o te ine whārite ki ngā tapatoru, inā koa ko ngā tapatoru tika, ko te koki kotahi he 90 nekehanga. Ko te whanaungatanga i waenga i ngā taha o te tapatoru me ōna koki e tohuhia ana e ngā mahi ine whārite taketake: sine (sin), cosine (cos), me te tangent (tan).
Ko ia o ēnei mahi e arotahi ana ki te ōwehenga i waenga i te roa o ngā taha o te tapatoru matau.
– Ko te sine (sin) o tētahi koki ko te ōwehenga i waenganui i te roa o te taha whakamuri me te roa o te hypotenuse.
– Ko te kosinī (cos) o tētahi koki ko te ōwehenga i waenganui i te roa o te taha tata me te roa o te porowhita.
– Ko te pātapa (tan) o tētahi koki ko te ōwehenga i waenganui i te roa o te taha whakamuri me te roa o te taha tata.
Hei mārama ake, me whakaaro tātou ki tētahi tapatoru matau me te koki θ:
""
|\
| \
o | \ h
| \
|—-|
a
""
Ngā Mōhiohio:
– ko o te roa o te taha whakarara, o te taha rerekē rānei o te koki θ.
– ko a te roa o te taha, te taha tata rānei o te koki θ.
– ko h te roa o te hypotenuse o te koki θ.
Nō reira, ka taea e tātou te tuhi i ngā mahi matua o te trigonometric penei:
– hara(θ) = o / h
– cos(θ) = a / h
– tan(θ) = o / a
2. He aha te hiranga o te Pānga-toru?
He whānuitia ngā whakamahinga o te ine whārite i roto i ngā momo mara o te mātauranga me te oranga o ia rā. Hei kaupapa matua, he maha ngā whakamahinga o te ine whārite i roto i:
– Ahupūngao: Te ine ngaru, te tātari kaha, me te nekehanga ā-wā.
– Tātai whetū: Te ine i te tawhiti i waenga i ngā whetū me te tatau i ngā porowhita o ngā aorangi.
– Hangarau: Te hoahoa i ngā whare, i ngā piriti, me ētahi atu hanganga.
– Whakatere: Te whakatau i te tūranga me te ahunga o tētahi kaipuke, o tētahi waka rererangi rānei.
3. Ngā Tuakiri Pātoru
Haunga ngā mahi taketake o te pākoki, kei reira anō ngā tuakiri pākoki he mea nui hei whakahaere i ngā tataunga:
– Tuakiri Pythagorean:
sin²(θ) + cos²(θ) = 1
– Te Tuakiri o te Tapeke me te Rerekētanga o ngā Koki:
hara(A + B) = hara(A)cos(B) + cos(A)sin(B)
cos(A + B) = cos(A)cos(B) – hara(A)sin(B)
– Tuakiri Takirua:
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) – hara²(θ)
Mā ēnei tuakiri ka taea te whakahaere me te whakangawari i ngā kīanga pāngarau.
4. He Kupu Whakataki ki ngā Wāhanga Koki
Ahakoa ko ngā nekehanga te waeine tino noa mō te ine i ngā koki, ka whakamahia anō hoki e te ine whārite te radian. Ko te porowhita katoa he 360 nekehanga, arā, 2π radian. Nō reira, ko te 1 radian he rite ki te 180/π nekehanga (tata ki te 57,2958 nekehanga).
Ka taea te whakawhiti i waenga i ngā nekehanga me ngā rātihana penei:
– Tahuri mai i ngā nekehanga ki ngā rātiana: ngā rātiana = ngā nekehanga × (π / 180)
– Tahuri mai i ngā rātiana ki ngā nekehanga: nekehanga = ngā rātiana × (180 / π)
5. Ngā Kauwhata Mahi Pāngatoru
Mā te tuhi kauwhata i ngā mahi pākoki ka taea te whakaatu i te āhua o tā rātou whakarerekētanga uara i runga i te koki. Anei tētahi tirohanga mō te āhua o tētahi kauwhata pēnei:
– Kauwhata Sine:
He ngaru tāruarua te kauwhata ngaru sine me te wā o te 2π. Kei te 1 te tihi (te uara mōrahi), ā, kei te -1 te awaawa (te uara iti rawa).
– Kauwhata Kōsina:
He rite te kauwhata kosinī ki te kauwhata sine engari ka tīmata mai i te uara mōrahi o te 1 i te koki 0. Pērā i te sine, ka tāruarua anō hoki tēnei kauwhata i ia 2π.
– Kauwhata Pānga:
E whakaatu ana te kauwhata pātata i tētahi rerekētanga nui ake, e pūmau ana i ia π. Ka eke tōna uara ki te mutunga kore ina whakatata atu ki te 90 nekehanga me te -90 nekehanga, ka hangaia he asymptote poutū.
6. Te Whakamahinga o te Ine Taurite: Te Whakaoti Rapanga
Me titiro tātou ki tētahi tauira o te whakamahinga o te ine pārōnaki i roto i te ao tūturu:
Tauira 1: Te Tātai i te Teitei o te Whare
Me kī kei te tū te tangata i te 50 mita mai i te turanga o te whare, ā, ka ine i te koki teitei ki te tihi o te whare he 30 nekehanga. E hia te teitei o te whare?
Mai i konei, ka whakamahia e mātou te mahi tangent:
tan(θ) = ritenga atu / tata
parauri (30°) = h / 50
h = 50 parauri (30°)
h ≈ 50 0.577 = 28.85 mita
Nō reira, ko te teitei o te whare he tata ki te 28.85 mita.
Tauira 2: Te Whakamahi i ngā Tuakiri Pātoru
Whakatauhia te uara o te sin(2θ) mēnā ko te sin(θ) = 3/5 me te cos(θ) = 4/5.
Ka whakamahia e mātou te tuakiri:
sin(2θ) = 2sin(θ)cos(θ)
hara(2θ) = 2 (3/5) (4/5)
hara(2θ) = 24/25 = 0.96
Nō reira, ko te sin(2θ) = 0.96.
Te Katinga
He taputapu pāngarau tino kaha, ngāwari hoki te ine pārōnaki. Mai i te mārama ki te nekehanga o ngā aorangi ki te hoahoa i ngā hanganga whare, ka whakaratohia e te ine pārōnaki he reo me te huarahi ki te tūhura me te mārama ki te ao e karapoti nei i a tātou. Mā te mōhio ki ēnei kaupapa matua o te ine pārōnaki, ka taea e te hunga tīmata te whakatuwhera i te kuaha ki te nui o ngā mātauranga matatau me ngā tono mahi i roto i te whānuitanga o ngā kaupapa ako.
