Ngā Tauira Pātai e Matapaki ana i te Whakamahinga o ngā Tauwehenga Pāngatoru
Ko te ine-ira he peka o te pāngarau e ako ana i ngā whanaungatanga i waenga i ngā taha me ngā koki o ngā tapatoru. He mea nui te mārama ki te ine-ira nā te mea he maha ngā whakamahinga i roto i ngā momo mara, tae atu ki te hoahoanga, te hangarau, te whetū, tae atu ki te whakamunatanga. Ka matapakihia e tēnei tuhinga ētahi tauira raruraru, ā, ka matapakihia hoki i roto i te horopaki o te whakamahi i ngā ōwehenga ine-ira.
Ngā Ariā Taketake o te Pāngatoru
I mua i te urunga atu ki ngā tauira rapanga, me arotake tātou i ētahi ariā taketake o te ine whārite. I roto i te tapatoru matau, e toru ngā mahi matua o te ine whārite e whakamahia ana: te sine, te cosine, me te tangent.
– Ko te sine (sin) o tētahi koki ko te ōwehenga o te roa o te taha e anga atu ana ki te koki ki te roa o te hypotenuse.
\[
\sin \theta = \frac{\text{taha whakamuri}}{\text{taupoki}}
\]
– Ko te kosinī (cos) o tētahi koki ko te ōwehenga o te roa o te taha e tata ana ki te koki ki te roa o te porowhita.
\[
\cos \theta = \frac{\text{taha tata}}{\text{taupoki}}
\]
– Ko te pātata (tan) o tētahi koki ko te ōwehenga o te roa o te taha e anga atu ana ki te koki ki te roa o te taha e tata ana ki te koki.
\[
\tan \theta = \frac{\text{taha mua}}{\text{taha taha}}
\]
Tauira Pātai 1: Te Tatau i te Teitei o te Pourewa
Pātai: E tū ana tētahi kaimātakitaki i te 50 mita te tawhiti mai i tētahi pourewa, ā, e ine ana i te koki teitei o te tihi o te pourewa kia 30 nekehanga. Tātaihia te teitei o te pourewa.
Otinga: Hei whakaoti i tēnei raruraru, ka taea e tātou te whakamahi i te mahi pātata i roto i te ine whārite. Nā te mea e mōhio ana tātou ki te koki o te teitei me te tawhiti whakapae mai i te kaimātakitaki ki te pourewa, ka taea e tātou te tuhi:
\[
\tan 30^\circ = \frac{\text{teitei o te pourewa}}{\text{tawhiti whakapae}}
\]
Whakakapia ngā uara e mōhiotia ana:
\[
\tan 30^\circ = \frac{h}{50}
\]
E mōhiotia ana ko \(\tan 30^\circ = \frac{1}{\sqrt{3}}\), nō reira:
\[
\frac{1}{\sqrt{3}} = \frac{h}{50}
\]
Kātahi ka taea te kimi i te teitei o te pourewa, \(h\), mā te whakarea i ngā taha e rua o te whārite mā te 50:
\[
h = 50 \cdot \frac{1}{\sqrt{3}} = \frac{50}{\sqrt{3}} \approx 28.87 \, \text{mita}
\]
Ko te teitei o te pourewa e 28.87 mita pea.
Tauira Pātai 2: Te Whakatau i te Tawhiti mā te Whakamahi i te Cosine
Pātai: E anga ana tētahi kaipuke ki te rawhiti mā te 10 kiromita, kātahi ka huri i te huarahi mā te 60 nekehanga ki te raki, ā, ka rere mā te 15 kiromita. Tātaihia te tawhiti mai i te tīmatanga ki te kaipuke.
Kōrero: Hei whakaoti i tēnei raruraru, ka taea e tātou te whakamahi i te ture cosine i roto i te trigonometry. Mēnā ka tuhia te haerenga o te kaipuke ki roto i tētahi pūnaha taunga, ka kitea he tapatoru me ngā taha o te 10 km me te 15 km, me te koki o te 60 nekehanga. Ka taea e tātou te whakamahi i te ture cosine hei kimi i te tawhiti i waenga i te pūwāhi tīmatanga o te kaipuke me tōna tūranga whakamutunga.
\[
c^2 = a^2 + b^2 – 2ab \cos C
\]
Dimana:
– \( a = 10 \)
– \( b = 15 \)
– \( C = 60^\pōtae \)
Whakakapia ngā uara e mōhiotia ana:
\[
c^2 = 10^2 + 15^2 – 2 \cdot 10 \cdot 15 \cdot \cos 60^\circ
\]
E mōhio ana tātou ko \(\cos 60^\circ = 0.5\), kātahi:
\[
c^2 = 100 + 225 – 2 \cdot 10 \cdot 15 \cdot 0.5
\]
\[
c^2 = 100 + 225 – 150
\]
\[
c^2 = 175
\]
\[
c = \sqrt{175} \approx 13.23 \, \text{km}
\]
Nō reira, ko te tawhiti mai i te tīmatanga ki te kaipuke he tata ki te 13.23 kiromita.
Tauira Pātai 3: Te Whakamahi i te Sine hei Whakatau i ngā Taha o te Tapatoru
Pātai: I roto i tētahi tapatoru, e 7 henemita te roa o ngā taha e rua, ā, e 10 henemita te whānui, ā, e 45 nekehanga te koki i waenganui i a rāua. Tātaihia te roa o te taha tuatoru o te tapatoru.
Kōrero: Ka taea e tātou te whakamahi i te ture o ngā sine hei whakaoti i tēnei raruraru. I roto i te ture o ngā sine, mō tētahi tapatoru me ngā taha \(a\), \(b\), me \(c\) me te koki \(C\) i waenganui i ngā taha \(a\) me \(b\):
\[
\frac{c}{\sin C} = \frac{a}{\sin A} = \frac{b}{\sin B}
\]
Heoi, i tēnei wā, ka taea e tātou te whakamahi tika i te ture o ngā cosine hei whakangāwari ake i ngā mahi. E mea ana te ture o ngā cosine:
\[
c^2 = a^2 + b^2 – 2ab \cos C
\]
Dimana:
– \( a = 7 \)
– \( b = 10 \)
– \( C = 45^\pōtae \)
\(\cos 45^\circ = \frac{\sqrt{2}}{2}\), kia:
\[
c^2 = 7^2 + 10^2 – 2 \cdot 7 \cdot 10 \cdot \frac{\sqrt{2}}{2}
\]
\[
c^2 = 49 + 100 – 70\sqrt{2}
\]
\[
c^2 = 149 – 70\sqrt{2}
\]
Te tatau i te uara o \( c \):
\[
c \approx \sqrt{149 – 70\sqrt{2}} \approx 5.97 \, \text{cm}
\]
Nō reira, ko te roa o te taha tuatoru o te tapatoru he 5.97 cm pea.
Whakamutunga
He taputapu tino whai hua te ine whārite mō te whakaoti rapanga maha e pā ana ki ngā tapatoru me ngā koki. Mā te mārama pai ki te sine, te cosine, te tangent, me ngā ture o te ine whārite, ka taea e tātou te whakaoti rapanga mahi maha. E matapakihia ana e tēnei tuhinga ētahi tauira o te whakamahi i ngā ōwehenga ine whārite, ā, ko tā mātou e tumanako nei ka āwhina i ngā kaipānui ki te whakawhānui ake i ō rātou māramatanga.
