Ngā Tauira Pātai e Matapaki ana i te Whakamahinga o ngā Tauwehenga Tautoru tan θ
Ko te ine whārite he peka o te pāngarau e pā ana ki ngā koki me ngā mahi koki i roto i ngā tapatoru. Ko tētahi ariā nui i roto i te ine whārite ko ngā ōwehenga ine whārite o ngā koki, pērā i te sine (sin), te cosine (cos), me te tangent (tan). I roto i tēnei tuhinga, ka arotahi tātou ki te tangent o tētahi koki kotahi θ, e tohuhia ana e te tan θ.
Te Whakamāramatanga o Tan θ
Ko te tānga o te koki θ i roto i te tapatoru matau ko te ōwehenga o te roa o te taha whakamuri o te koki θ ki te roa o te taha tata o te koki θ. Mā te pāngarau, ka whakaatuhia te tan θ penei:
\[ \tan \theta = \frac{\text{te taha whakarara o te koki θ}}{\text{te taha tata o te koki θ}} \]
Hei mārama ake i tēnei ariā, ka tirohia e tātou ētahi tauira rapanga, ka matapakihia hoki ngā whakamahinga o te tan θ.
Tauira Pātai 1: Te tātai i a Tan θ
Homai he tapatoru matau me te koki θ kei te pūwāhi A, ko te roa o te taha whakarara o te koki θ he 3 cm, ā, ko te roa o te taha tata o te koki θ he 4 cm. Tātaihia te tan θ.
Otinga:
Mai i ngā raruraru o runga ake nei, e mōhio ana tātou:
– Ko te taha whakarara o te koki θ (kei te ritenga kē) = 3 cm
– Te taha tata o te koki θ = 4 cm
Mā te whakamahi i te whakamāramatanga o te tan θ, ka tatauhia e mātou:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
\[ \tan \theta = \frac{3}{4} \]
Nō reira, tan θ = 0.75.
I te taha āhuahanga, ko te tikanga o tēnei, mō te koki θ i roto i te tapatoru, ko te ōwehenga o te roa o te taha whakarara ki te roa o te taha tata ko 0.75.
Tauira 2: Te Whakamahi i te Tan θ hei Tātai i te Roa Taha
E whakawhirinaki ana tētahi arawhata ki te pakitara i te koki teitei θ o te 30 nekehanga. Ko te tawhiti mai i te waewae o te arawhata ki te pakitara he 5 mita. Kia pēhea te roa o te whakawhirinaki o te arawhata ki te pakitara?
Otinga:
Ko te taahiraa tuatahi, ka maumahara tātou ki te whakamāramatanga o te tan θ:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
I roto i te horopaki o tēnei raruraru:
– θ = 30 nekehanga
– te tawhiti mai i te take o te arawhata ki te pakitara) = 5 mita
– te ritenga atu (te teitei o te arawhata ki te pakitara) = ???
Ka tatau tuatahi mātou\text{opposite)):
\[ \tan 30^\circ = \frac{\text{opposite}}{5} \]
E mōhio ana tātou mai i te ripanga pātoru:
\[ \tan 30^\circ = \frac{\sqrt{3}}{3} \]
Nā reira:
\[ \frac{\sqrt{3}}{3} = \frac{\text{opposite}}{5} \]
Whakareatia ngā taha e rua ki te 5:
\[ \text{opposite} = 5 \cdot \frac{\sqrt{3}}{3} = \frac{5\sqrt{3}}{3} \]
Ko te ritenga atu (te teitei o te arawhata ki te pakitara) ko:
\[ \frac{5\sqrt{3}}{3} \approx 2.89 \text{ mita} \]
Nō reira, e 5 mita te roa o te arawhata.
Tauira 3: Te Tātai i ngā Koki mā te whakamahi i te Tan θ
E 12 mita te roa o te atarangi o tētahi pourewa. Mēnā e 8 mita te teitei o te pourewa, he aha te koki teitei o te rā θ?
Otinga:
I roto i tēnei raruraru, ka hoatu ki a tātou:
– Teitei o te pourewa (kei te ritenga atu) = 8 mita
– Te roa o te atarangi (e tata ana) = 12 mita
Ka whakamahia e mātou te whakamāramatanga o te tan θ hei kimi i te θ:
\[ \tan \theta = \frac{8}{12} = \frac{2}{3} \]
Nā, ka kitea e tātou a θ me te whārite:
\[ \theta = \tan^{-1} \left(\frac{2}{3}\right) \]
Mā te tirotiro i tētahi ripanga, i tētahi tātaitai rānei hei whakatau i te uara o te tātaitai whakamuri, ka kitea e tātou:
\[ \theta \tata ki te 33.69^\circ \]
Nō reira, ko te koki teitei o te rā he tata ki te 33.69 nekehanga.
Tauira 4: Te Whakamahi i te Tan θ ki ngā Hiahia o te Ao Tūturu
Kua tāutahia he whakaata mārama e mau ana ki runga i tētahi pou 4-mita te teitei i runga ake i tētahi motuka. Ki te hiahia koe ki te tāuta i tētahi haona e kitea ana i te koki 45-tohu mai i te whenua, tatauhia te tawhiti roa rawa atu e kitea tonu ai te haona.
Otinga:
Mai i te pātai, e mōhiotia ana:
– Teitei o te pou (kei te ritenga atu) = 4 mita
– Koki θ = 45 nekehanga
E ai ki te whakamāramatanga o te tan θ:
\[ \tan 45^\circ = \frac{\text{opposite}}{\text{adjacent}} \]
E mōhio ana tātou ko \(\tan 45^\circ = 1\), nō reira:
\[ 1 = \frac{4}{\text{adjacent}} \]
Nā reira:
\[ \text{adjacent} = 4 \text{ mita} \]
Nō reira, ko te tawhiti tawhiti rawa atu e kitea ai te haona he 4 mita.
Whakamutunga
Mai i ngā tauira i runga ake nei, ka kite tātou he ariā tino whai hua te pātaki o te koki θ (\(\tan \theta\)), ā, he whānuitia ngā tono mahi, mai i te whakaoti rapanga ngāwari i roto i te pāngarau ki tōna tono i roto i ngā mahi o ia rā, pērā i te hanga me te whakatere. Mā te mārama pai ki tēnei ariā ka āwhina i te whakaoti rapanga maha e pā ana ki te whakatairite i te roa o ngā taha o te tapatoru.
I te nuinga o te wā, ko te tan θ, hei wāhanga o te ine whārite, ehara i te mea he kaupapa nui noa iho i roto i te mātauranga ōkawa engari he taputapu tino whai hua hoki i roto i ngā āhuatanga maha o te ao tūturu. Ko te tumanako, ka whakaratohia e tēnei tuhinga he tirohanga whānui mārama me te hōhonu mō te whakamahi i te tan θ hei whakaoti rapanga e pā ana.
