Tauira Pātai me te Kōrero mō tētahi Momo Tauwehenga Tautoru: tan θ
Ko te ine whārite he peka o te pāngarau e ako ana i te whanaungatanga i waenga i ngā koki me ngā roa taha i roto i ngā tapatoru. Ko tētahi ōwehenga ine whārite e kōrerohia pinepinetia ana ko te tāne whārite (tan). I roto i tēnei tuhinga, ka arotahi tātou ki te whakamahi i te ōwehenga tan i roto i ngā momo raruraru, ā, ka matapakihia ētahi tauira e pā ana ki te tan θ.
Te whakamāramatanga o te tan θ
Ko te tānga o te koki θ e tautuhia ana ko te ōwehenga o te roa o te taha whakamuri ki te roa o te taha tata i roto i te tapatoru matau. Mā te pāngarau, ka tuhia tēnei penei:
\[ \tan θ = \frac{\text{taha ritenga}}{\text{taha tata}} \]
I roto i te porowhita kotahi, ka taea hoki te whakamārama i te tan hei ōwehenga i waenga i te taunga y (taha mua) me te taunga x (taha taha) o tētahi pūwāhi i runga i te porowhita e kotahi te wae te tawhiti atu i te pokapū.
Te mahi tan i roto i te Pāngarau me te Ahupūngao
Ko te ine whārite, inā koa te mahi tan, e whakamahia ana i roto i ngā momo mahi pāngarau me ngā mahi ā-tinana. Hei tauira, i roto i te ahupūngao matarohia, ka whakamahia te mahi tan i roto i te tātari i te nekehanga pere, ā, i roto i te hangarau, ka whakamahia hei tatau i te koki o te piko, te pikinga rānei o te mata.
Ngā Pātai Tauira me te Kōrero
Anei ētahi tauira pātai me ā rātou matapakinga hei mārama ake i te whakamahinga o te tan θ.
Pātai 1: Te tatau i te tan θ o tētahi tapatoru matau
Hoatu: Ko te roa o te taha o mua o te tapatoru matau e anga atu ana ki te koki θ he 4 cm, ā, ko te roa o te taha e tata ana ki te koki θ he 3 cm. Tātaihia te uara o te tan θ.
Kōrero:
Whakamahia te whakamāramatanga o te parauri:
\[ \tan θ = \frac{\text{taha mua}}{\text{taha taha}} \]
Whakakapia ngā uara e mōhiotia ana:
\[ \tan θ = \frac{4}{3} \]
Nō reira, ko te uara o te tan θ ko \( \frac{4}{3} \).
Pātai 2: Te whakatau i te roa o tētahi taha mā te whakamahi i te tan θ
Hoatu: E mōhiotia ana ko te tapatoru matau me te koki θ ko te tan θ = 0.75. Ko te roa o te taha e tata ana ki te koki θ he 8 cm. Tātaihia te roa o te taha whakamuri e whakamuri ana i te koki θ.
Kōrero:
Whakamahia te whakamāramatanga o te parauri hei kimi i te roa o te taha whakarara:
\[ \tan θ = \frac{\text{taha mua}}{\text{taha taha}} \]
\[ 0.75 = \frac{\text{taha mua}}{8} \]
Whakareatia ngā taha e rua ki te 8 hei whakaoti i te whārite.
\[ \text{taha mua} = 0.75 \times 8 \]
\[ \text{taha mua} = 6 henemita \]
Nō reira, ko te roa o te taha o mua he 6 cm.
Pātai 3: Te tatau i te koki θ mēnā e mōhiotia ana te tan θ
Hoatu: E mōhiotia ana ko te tan θ = 1 te koki o tētahi tapatoru matau. Whakaingoatia te koki θ.
Kōrero:
Ko te tae parauri o tētahi koki he ōrite ki te 1 ina ōrite te roa o te taha whakamuri me te taha tata. I roto i te ine whārite taketake, ka puta tēnei i te koki 45°.
Nō reira, ko te uara o θ he 45°.
Pātai 4: Te whakamahi i te Tan θ i roto i ngā rapanga pāngarau
Hoatu: E herea ana tētahi taura mai i te tihi o tētahi pou e 15 mita te teitei ki tētahi pūwāhi i runga i te whenua e 20 mita te tawhiti mai i te pūtake o te pou. Tātaihia te tan θ, ko θ te koki i hangaia e te taura me te pou.
Kōrero:
Whakamahia te whakamāramatanga o te parauri:
\[ \tan θ = \frac{\text{taha mua (teitei o te pou)}}{\text{taha taha (tawhiti whakapae)}} \]
\[ \tan θ = \frac{15}{20} \]
Whakangāwaritia te hautau:
\[ \tan θ = \frac{3}{4} \]
Nō reira, ko te uara o te tan θ ko \( \frac{3}{4} \).
Pātai 5: Te whakatau i te teitei mai i te tawhiti me te koki o te piko
Hoatu: E tū ana tētahi kaimātakitaki i te 100 mita mai i tētahi whare teitei. Ko te tan θ o te tirohanga mai i te tūranga o te kaimātakitaki ki te tihi o te whare ko \(\tan 30^\circ\). Whakatauhia te teitei o te whare.
Kōrero:
E mōhiotia ana ko \(\tan 30^\circ = \frac{1}{\sqrt{3}}\).
\[ \tan θ = \frac{\text{taha mua (teitei o te whare)}}{\text{taha taha (tawhiti)} } \]
Monohia ngā uara e mōhiotia ana ki roto i te whārite
\[ \frac{1}{\sqrt{3}} = \frac{\text{teitei o te whare}}{100} \]
Whakareatia ngā taha e rua ki te 100 kia wehea ai te teitei.
\[ \text{teitei o te whare} = \frac{100}{\sqrt{3}} \]
\[ \text{teitei o te whare} = \frac{100 \times \sqrt{3}}{3} \]
\[ \text{teitei o te whare} ≈ 57.73 \text{ mita} \]
Nō reira, ko te teitei o te whare he tata ki te 57.73 mita.
Pātai 6: Te whakatau i te koki mai i te teitei me te tawhiti
Hoatu: E mōhio ana koe ko te teitei o te pourewa he 50 mita, ā, ko te tawhiti whakapae mai i te pūwāhi tirotiro ki raro o te pourewa he 70 mita. Whakatauhia te koki teitei ki te tihi o te pourewa.
Kōrero:
\[ \tan θ = \frac{\text{teitei o te pourewa}}{\text{tawhiti whakapae}} \]
\[ \tan θ = \frac{50}{70} \]
\[ \tan θ = \frac{5}{7} \]
Hei kimi i a θ, ka whakamahia e mātou te mahi pānihi whakamuri (tan⁻¹) te arctan rānei.
\[ θ = \tan⁻¹ (\frac{5}{7}) \]
Mā te whakamahi i tētahi tātaitai, i tētahi ripanga ira-whārite rānei, ka kitea te uara o θ.
\[ θ ≈ 35.54° \]
Nō reira, ko te koki teitei ki te tihi o te pourewa he tata ki te 35.54°.
Whakamutunga
He taputapu kaha te ine whārite i roto i ngā mara maha o te pūtaiao. Hei tauira, he ōwehenga māmā engari kaha te tangent ka taea te whakamahi hei whakaoti rapanga maha e pā ana ki ngā koki me ngā roa taha. Mā te mārama ki tōna whakamāramatanga me te whakamahinga, ka taea e tātou te whakaoti i te whānuitanga o ngā rapanga āhuahanga me te ahupūngao. Mā te whakaharatau i ngā rapanga pēnei i te tauira i runga ake nei, ka taea e tātou te whakamahi i te tan θ i roto i ngā tataunga o ia rā.
