Ngā Tauwehenga Pātoru e Toru
Ko te ine whārite he peka o te pāngarau e ako ana i te whanaungatanga i waenga i ngā koki me ngā roa taha o te tapatoru. I roto i te oranga o ia rā, he maha ngā wā ka tūtaki tātou ki ngā ariā ine whārite, i roto i te hangarau, i te ahupūngao, i te whetū, tae atu ki ngā tono māmā pēnei i te whakatauira, te pakiwaituhi rānei. Ko tētahi o ngā huānga matua o te ine whārite e whakamahia pinepinetia ana ko ngā mahi ōwehenga ine whārite, e toru ngā mahi matua: sine (sin), cosine (cos), me te tangent (tan).
Sine (Hina)
Ko te sine tētahi o ngā mahi matua o te trigonometric. Mō te koki $θ$ i roto i te tapatoru matau, ko te sine te ōwehenga i waenga i te roa o te taha e anga atu ana ki te koki $θ$ (te taha e anga atu ana) me te roa o te hypotenuse.
\[
\text{sin}(θ) = \frac{\text{taha horapa}}{\text{hypotenuse}}
\]
Ariā me te Whakamahinga
He mea nui te mahi sine i roto i te whānuitanga o ngā tono. Hei tauira, i roto i te ahupūngao, ka whakamahia te sine i roto i ngā tauira ngaru sinusoidal e whakaahua ana i te mārama, te oro, me ētahi atu ngaru hikohiko. I roto i te hangarau, ka whakamahia te sine mō te tātari wiri me te hoahoa ara iahiko AC.
Whakaarohia te piki maunga i tētahi koki. Mēnā e mōhio ana tātou ki te roa o te pari o te ara e pikihia ana e tātou, ā, e hiahia ana hoki tātou ki te mōhio ki te teitei kua tae atu tātou, ka taea e tātou te whakamahi i te mahi sine, me te mōhio ki te koki o te teitei me te roa o te ara.
Kōsine (Kos)
Ko te kosinī tētahi atu mahi whaihua rite tonu te hiranga. Mō te koki $θ$ i roto i te tapatoru matau, ko te kosinī te ōwehenga o te roa o te taha e tata ana ki te koki $θ$ (te taha e tata ana) ki te roa o te porowhita.
\[
\text{cos}(θ) = \frac{\text{taha tata}}{\text{taumata}}
\]
Ariā me te Whakamahinga
He maha ngā whakamahinga o te kosinī, pērā i te sine, i roto i ngā momo mara. I roto i te ahupūngao, ka whakamahia te kosinī hei ine i te wāhanga whakapae o tētahi whārite. Hei tauira, ki te whiua e tātou he pōro ki tētahi koki ki te whenua, ā, e hiahia ana tātou ki te mōhio ki te tawhiti o tana haere whakapae, ka taea e tātou te whakamahi i te mahi kosinī.
Ko tētahi atu tauira kei roto i te hoahoa whakairoiro me te pakiwaituhi rorohiko. I konei, ka whakamahia te mahi cosine mō te hurihanga o ngā mea toru-ahu me ngā panonitanga taunga.
Pātene (Tan)
Ko te pāne te tuatoru o ngā mahi pāne e pā tata ana ki te sine me te cosine. Mō te koki $θ$ i roto i te tapatoru matau, ko te pāne te ōwehenga i waenga i te roa o te taha e anga atu ana ki te koki $θ$ (te taha ke) me te roa o te taha e tata ana ki te koki $θ$ (te taha e tata ana).
\[
\text{tan}(θ) = \frac{\text{taha mua}}{\text{taha taha}}
\]
Ariā me te Whakamahinga
He maha ngā wā ka whakamahia ngā tātaitanga hei tatau i te teitei o ngā mea, o ngā pari rānei, he mea tino whai hua tēnei i roto i te whenua me te hangarau ā-iwi. Hei tauira, ko te kairūri whenua e hiahia ana ki te mōhio ki te teitei o tētahi whare, me ine noa i te tawhiti mai i a ia anō ki te whare me te koki teitei o tana tirohanga. Mā te whakamahi i te tātaitanga o taua koki me te tawhiti e mōhiotia ana, ka taea te tatau ngāwari i te teitei o te whare.
Hei tāpiritanga, ka whakamahia hoki te pātata i roto i te whakatere hei whakatau i te ahunga me te azimuth. I roto i te rererangi, ka whakamahia te pātata hei tatau i ngā ara rere whai hua, me te whakaaro ki te hau me ētahi atu āhuatanga o waho.
Te Hononga i waenganui i te Tokotoru
He hononga tata tā te sine, te cosine, me te tangent i roto i te pāngarau. Ko tētahi o ngā tuakiri trigonometric rongonui ko te tuakiri Pythagorean, e hono ana i te sin me te cos i roto i tētahi whārite:
\[
\text{sin}^2(θ) + \text{cos}^2(θ) = 1
\]
Kātahi, he whanaungatanga tika kei waenganui i te tangent, te sine, me te cosine:
\[
\text{tan}(θ) = \frac{\text{sin}(θ)}{\text{cos}(θ)}
\]
E whakaatu ana i te hononga o ēnei mahi e toru tetahi ki tetahi i roto i ngā horopaki pākokiora rerekē.
Ngā Tikanga Whakamaumahara me te Mahara
He uaua pea te maumahara i ngā uara me ngā whanaungatanga i waenga i ngā mahi pārōnaki. Ko tētahi tikanga maumahara rongonui ko te whakamahi i ngā ripanga pārōnaki, i ngā taputapu maumahara rānei. Ko tētahi maumahara rongonui ko te SOH-CAH-TOA, ko tōna tikanga:
– SOH: Sine = Te ritenga kē / Te taha whakarara
– CAH: Kosine = Tata / Āpiti
– TOA: Pātata = Whakahē / Tata
Mā tēnei tikanga ka āwhina i ngā ākonga ki te maumahara ki ngā whakamāramatanga taketake o ia pānga pākoki.
Te Whakatere i roto i ngā Whakairoiro
He mea nui anō hoki ngā kauwhata o ngā mahi pākoki hei whakarato i te tirohanga tirohanga. Hei tauira, ko ngā kauwhata o te sine me te cosine he ngaru tāruarua me te wā o te $2π$. Ka tīmata te ngaru sine i te 0, ka piki ki te 1, ka heke ki te -1, ka hoki anō ki te 0, ka hanga he huringa ōrite.
Ko te kauwhata pātata, i tētahi atu taha, he rerekē te whanonga nā te mea kāore i tautuhia i ngā koki $π/2$, $3π/2$, me ērā atu. He asymptotes kei te kauwhata pātata e whakatata atu ana ki te mutunga kore pai, kino rānei i ēnei koki.
Mā te mārama ki ēnei kauwhata, ka māmā ake tā tātou tiro me te mārama ki te whanonga o ngā pānga pākoki i roto i ngā horopaki rerekē.
Ngā Wero me te Ngahau o te Ako i te Pāngarau
Ahakoa te āhua uaua o te ine pārāki, mā te mārama ki ngā mahi pārāki—te sine, te cosine, me te tangent—ka taea te whakatuwhera i ngā tatau ki ngā mara maha o te pūtaiao, ā, ka taea hoki te whakamahi i roto i te oranga o ia rā. Ko te mea nui ko te mārama ki ngā whakamāramatanga taketake, ngā whanaungatanga i waenga i ia mahi, me pēhea te whakamahi hei whakaoti rapanga o te ao tūturu.
Mā te whakaharatau auau me te whakamahi i ngā taputapu pēnei i ngā ripanga me ngā kauwhata ka tino āwhina i te mārama ki ēnei ariā. I tēnei ao matihiko, he maha ngā taupānga me ngā taputapu ipurangi ka taea te whakahaere i te tukanga ako, pēnei i ngā tātaitai ine whārite me ngā whakaaturanga whakairoiro tauwhitiwhiti.
Whakamutunga
Ko te sine, te cosine, me te tangent ngā ine-taurite e kore e taea te wehe. Mā te mārama ki te whanaungatanga i waenga i ngā koki me ngā roa taha i roto i ngā tapatoru, ka mārama hoki tātou ki te mahi a te ao e karapoti nei i a tātou—mai i te nekehanga o ngā aorangi, ki ngā wiri o ngā ngaru, ki te hanganga o ngā whare. Ka ako mai te ine-taurite ehara te pāngarau i te mea mō ngā tau anake, engari mō te māramatanga hohonu ake o te ao whānui. Nō reira, ko te mōhio ki ēnei ariā taketake he taahiraa nui ki te mōhio ki ngā mara whānui o te mātauranga.
