Ngā Tauira Pātai e Matapaki ana i ngā Tauwehenga Pāngatoru
Ko te ine whārite he peka o te pāngarau e pā ana ki te whanaungatanga i waenga i ngā roa me ngā koki i roto i ngā tapatoru. He mahi nui tā ngā ōwehenga ine whārite i roto i ngā momo mara pēnei i te hangarau, te ahupūngao, te whetū, tae noa ki te oranga o ia rā. Ka arotakehia e tēnei tuhinga ētahi tauira rapanga me ā rātou otinga hei āwhina i a koe ki te mārama ki ngā ariā taketake o ngā ōwehenga ine whārite.
Ngā ōwehenga pātoru i roto i ngā tapatoru matau
Me tīmata mā te mārama ki ngā ōwehenga taketake o te tapatoru matau. Ko ēnei ōwehenga e mōhiotia ana ko te sine (sin), te cosine (cos), me te tangent (tan). I roto i te tapatoru matau, e toru ngā taha nui me tautuhi e tātou:
1. Taha Whakahē: Ko te taha e anga atu ana ki te koki e whakaarohia ana e tātou.
2. Taha Pātata: Ko te taha e tata ana ki te koki e whakaarohia ana e tātou.
3. Hipotenusa: Ko te taha roa rawa atu o te tapatoru matau e anga atu ana ki te koki matau.
Tauira Pātai 1
Ki te hoatu he tapatoru matau me te koki θ, ko te taha whakarara ki te koki θ he 3 waeine te roa, ko te taha tata he 4 waeine te roa, ā, ko te hypotenuse he 5 waeine te roa. Kimihia ngā uara o te sin, cos, me te tan o te koki θ.
Otinga:
Hei kimi i ngā uara sine, cosine, me te tangent o te koki θ, ka whakamahia e mātou ngā tātai ōwehenga trigonometric taketake:
– Sine (hara) θ = Taha Mua / Hypotenuse
\[ \sin θ = \frac{3}{5} \]
– Kosine (cos) θ = Taha taha / Hipotenuse
\[ \cos θ = \frac{4}{5} \]
– Pānga (pango) θ = Taha o Mua / Taha
\[ \tan θ = \frac{3}{4} \]
Ngā Hononga Pānga-toru o Ētahi Atu Koki
Ka taea hoki te whakamahi i ngā ōwehenga pākoki ki ētahi atu koki i roto i ngā momo tapatoru rerekē, tae atu ki ngā tapatoru ōrite me ngā tapatoru auau. He mea nui kia mārama ki ngā ture taketake pēnei i te Ture Sine me te Ture Cosine, e pā ana ki ngā tapatoru ehara i te tapatoru tika.
Tauira Pātai 2
Homai he tapatoru ABC me ngā taha a = 7 cm, b = 24 cm, me te koki C = 90°. Kimihia te roa o te taha c me ngā uara o ngā koki A me B.
Otinga:
Nā te mea he koki matau te koki C, ka taea e tātou te whakamahi i te Pythagorean Theorem hei kimi i te roa o te taha c (te hypotenuse):
\[ c = \sqrt{a^2 + b^2} \]
\[ c = \sqrt{7^2 + 24^2} \]
\[ c = \sqrt{49 + 576} \]
\[ c = \sqrt{625} \]
\[ c = 25 \, \kuputuhi{cm} \]
Muri iho, hei whakatau i te uara o ngā koki A me B, ka taea e tātou te whakamahi i ngā ōwehenga pākoki taketake.
Mō te koki A, ka whakamahia e mātou te kosinī:
\[ \cos A = \frac{\text{taha tata}}{\text{taupoki}} = \frac{24}{25} \]
\[ A = \cos^{-1} \left(\frac{24}{25}\right) \]
Mō te koki B, ka whakamahia te sine:
\[ \sin B = \frac{\text{taha whakamuri}}{\text{taumata}} = \frac{24}{25} \]
\[ B = \sin^{-1} \left(\frac{24}{25}\right) \]
Nā te mea i roto i te tapatoru matau, me 90° te tapeke o ngā koki ehara i te koki matau:
\[ A + B = 90° \]
\[ A = 90° – B \]
Ture Sine
Ko te ture sine e whai ake nei:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Tauira Pātai 3
Homai he tapatoru ko te taha a = 8 cm, te taha b = 15 cm, me te koki C = 60°. Kimihia te koki A me te roa o te taha c mā te whakamahi i te Ture Sines.
Otinga:
Tuatahi, whakamahia te Ture o ngā Sines hei kimi i tētahi o ngā koki kē atu (hei tauira, te koki A):
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
\[ \frac{8}{\sin A} = \frac{c}{\sin 60°} \]
Me tatau tuatahi tātou i te uara o c. Nā te mea he 60° te koki C, ka taea e tātou te whakamahi i te Ture Cosine:
\[ c^2 = a^2 + b^2 – 2ab \cos(C) \]
\[ c^2 = 8^2 + 15^2 – 2 \cdot 8 \cdot 15 \cdot \cos(60°) \]
\[ c^2 = 64 + 225 – 240 \cdot 0.5 \]
\[ c^2 = 289 – 120 \]
\[ c^2 = 169 \]
\[ c = 13 \, \kuputuhi{cm} \]
Inaianei, ka kitea e tātou te koki A:
\[ \frac{8}{\sin A} = \frac{13}{\sin 60°} \]
Mai i te mea ko \(\sin 60° = \sqrt{3}/2\):
\[ \frac{8}{\sin A} = \frac{13}{\sqrt{3}/2} \]
\[ \frac{8}{\sin A} = \frac{26}{\sqrt{3}} \]
\[ 8 \cdot \sqrt{3} = 26 \sin A \]
\[ \sin A = \frac{8 \sqrt{3}}{26} \]
\[ \sin A = \frac{4 \sqrt{3}}{13} \]
Hei whakamutunga, mā te whakamahi i te sine whakamuri:
\[ A = \sin^{-1}\left(\frac{4 \sqrt{3}}{13}\right) \]
Ture Kōsina
E mea ana te ture cosine:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
Tauira Pātai 4
Homai he tapatoru ko te taha a = 9 cm, te taha b = 12 cm, me te taha c = 15 cm. Kimihia te koki C.
Otinga:
Hei kimi i te koki C, ka whakamahia e mātou te Ture Cosine:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
\[ 15^2 = 9^2 + 12^2 – 2 \cdot 9 \cdot 12 \cdot \cos C \]
\[ 225 = 81 + 144 – 216 \cos C \]
\[ 225 = 225 – 216 \cos C \]
\[ 0 = -216 \cos C \]
\[ \cos C = 0 \]
Mai i konei ka mōhio tātou ko te koki C he 90° nā te mea ko cos 90° = 0.
Anei ētahi tauira rapanga hei āwhina i a koe ki te mārama ki ngā ōwehenga pākoki. Mā te whakaharatau tonu ki ngā momo rapanga ka tino āwhina i a koe ki te whakapakari i tō māramatanga ki ēnei ariā. Ko te tumanako kua whai hua tēnei tuhinga ki a koe ki te ako!
