Ngā Tauira o ngā Pātai me ngā Kōrero mō ngā Koki Motuhake i roto i ngā Tauwehenga Pāngatoru
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 tētahi ariā nui i roto i te ine whārite ko te whakamahinga o ngā koki motuhake hei mārama ki ngā ōwehenga ine whārite. Ko ngā koki motuhake e whakamahia whānuitia ana ko te 0°, 30°, 45°, 60°, me te 90°. Ka whakamāramahia e tēnei tuhinga ngā tauira me te matapaki i ngā koki motuhake i roto i ngā ōwehenga ine whārite.
He Kupu Whakataki ki ngā Koki Motuhake
Ka whiwhihia ngā koki motuhake mā te tātari i ngā tapatoru motuhake, pērā i ngā tapatoru rite-waewae me ngā tapatoru ōrite-taha. Anei ngā uara tapatoru taketake mō ngā koki motuhake hei maumahara:
| Koki (θ) | Hara(θ) | Cos(θ) | Tan(θ) |
|———–|——–|——–|——–|——–|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | – |
Mā te mōhio ki ēnei uara taketake, ka taea e tātou te whakaoti rapanga maha e pā ana ki ngā ōwehenga pākoki o ngā koki motuhake.
Ngā Pātai Tauira me te Kōrero
Me titiro tātou ki ētahi tauira pātai me ā rātou matapakinga:
Tauira Pātai 1
Pātai:
Tātaihia te uara o \( \sin(30°) + \cos(60°) \).
Kōrero:
Ka whakamahia e mātou ngā uara taketake o te ine ine koki motuhake.
\[
\sin(30°) = \frac{1}{2}
\]
\[
\cos(60°) = \frac{1}{2}
\]
Nō reira,
\[
\sin(30°) + \cos(60°) = \frac{1}{2} + \frac{1}{2} = 1
\]
Na, \( \sin(30°) + \cos(60°) = 1 \).
Tauira Pātai 2
Pātai:
Whakatauhia te uara o \( \tan(45°) \times \cos(45°) \).
Kōrero:
Ka whakamahia e mātou ngā uara mai i te ripanga koki motuhake.
\[
tan(45°) = 1
\]
\[
\cos(45°) = \frac{\sqrt{2}}{2}
\]
Nō reira,
\[
\tan(45°) \times \cos(45°) = 1 \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}
\]
Nō reira, \( \tan(45°) \times \cos(45°) = \frac{\sqrt{2}}{2} \).
Tauira Pātai 3
Pātai:
Mena ko \( \sin(θ) = \cos(θ) \), whakatauhia te uara o \( θ \) i te awhe 0° ki te 90°.
Kōrero:
Mai i ngā whanaungatanga taketake o te ine whārite:
\[
\sin(θ) = \cos(θ)
\]
Ko te tikanga tēnei ko \( \tan(θ) = 1 \).
Ko te uara o \( θ \) e tutuki ana i te whārite \( \tan(θ) = 1 \) he 45°.
Nō reira, \( θ = 45° \).
Tauira Pātai 4
Pātai:
Tātaihia te uara o \( \frac{\sin(30°)}{\cos(60°)} \).
Kōrero:
Ka whakamahia e mātou ngā uara mai i te ripanga koki motuhake.
\[
\sin(30°) = \frac{1}{2}
\]
\[
\cos(60°) = \frac{1}{2}
\]
Nō reira,
\[
\frac{\sin(30°)}{\cos(60°)} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1
\]
Na, \( \frac{\sin(30°)}{\cos(60°)} = 1 \).
Tauira Pātai 5
Pātai:
Whakatauhia te uara o \( \cos(30°) \times \tan(60°) \).
Kōrero:
Ka whakamahia e mātou ngā uara mai i te ripanga koki motuhake.
\[
\cos(30°) = \frac{\sqrt{3}}{2}
\]
\[
\tan(60°) = \sqrt{3}
\]
Nō reira,
\[
\cos(30°) \times \tan(60°) = \frac{\sqrt{3}}{2} \times \sqrt{3} = \frac{3}{2}
\]
Nō reira, \( \cos(30°) \times \tan(60°) = \frac{3}{2} \).
Tauira Pātai 6
Pātai:
Kimihia te uara o \( 2 \sin(45°) \cos(45°) \).
Kōrero:
Ka whakamahia e mātou ngā uara mai i te ripanga koki motuhake.
\[
\sin(45°) = \frac{\sqrt{2}}{2}
\]
\[
\cos(45°) = \frac{\sqrt{2}}{2}
\]
Nō reira,
\[
2 \sin(45°) \cos(45°) = 2 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} = 2 \times \frac{2}{4} = 1
\]
Na, \( 2 \sin(45°) \cos(45°) = 1 \).
Tauira Pātai 7
Pātai:
Whakatauhia te uara o \( \csc(30°) \).
Kōrero:
\( \csc(θ) \) ko te kōaro o te \( \sin(θ) \).
\[
\sin(30°) = \frac{1}{2}
\]
Nō reira,
\[
\csc(30°) = \frac{1}{\sin(30°)} = \frac{1}{\frac{1}{2}} = 2
\]
Nō reira, \( \csc(30°) = 2 \).
Tauira Pātai 8
Pātai:
Tātaihia te uara o \( \cot(60°) \).
Kōrero:
Ko \( \cot(θ) \) te kōaro o te \( \tan(θ) \).
\[
\tan(60°) = \sqrt{3}
\]
Nō reira,
\[
\cot(60°) = \frac{1}{\tan(60°)} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\]
Nō reira, \( \cot(60°) = \frac{\sqrt{3}}{3} \).
Tauira Pātai 9
Pātai:
Mena he koki a \( \theta \) ko tōna uara pātoru ko \( \sin(\theta) = \cos(45°) \), kimihia te uara o \( \theta \) i te awhe 0° ki te 90°.
Kōrero:
Mai i te ripanga koki motuhake:
\[
\cos(45°) = \frac{\sqrt{2}}{2}
\]
Nō reira,
\[
\sin(\theta) = \frac{\sqrt{2}}{2}
\]
E mōhiotia ana,
\[
\sin(45°) = \frac{\sqrt{2}}{2}
\]
Nō reira, \( \theta = 45° \).
Whakamutunga
He mea nui te mōhio ki ngā koki motuhake me ngā uara taketake o te ira-whakakotahi hei mārama ki ngā ariā o te ira-whakakotahi me te whakaoti rapanga pāngarau. Mā te mahi tika, ka māmā ake te maumahara i te ripanga koki motuhake, ā, ka tere ake, ka whai hua ake hoki te whakaoti rapanga ira-whakakotahi.
Hei whakamutunga, ka whakaatuhia e tēnei tuhinga ētahi tauira rapanga me ngā kōrero e pā ana ki ngā koki motuhake, hei āwhina i a koe ki te mārama ki te whakamahi i ngā uara pārōno o ngā koki motuhake i roto i te mahi. Ko te tumanako kua whai hua tēnei tuhinga i roto i tō ako!
