Nā nīnau hoʻohālike e kūkākūkā ana i nā lakio Trigonometric ma nā Pyramids
Pendahuluan
ʻO ke kumu, ʻo ka trigonometry kahi lālā o ka makemakika e aʻo ana i nā pilina ma waena o nā kihi a me nā ʻaoʻao o nā huinakolu. He mea koʻikoʻi nā lakio trigonometric i nā ʻano like ʻole e like me ke kino, ka ʻenekinia, a me ke kuhikuhipuʻuone. ʻO kekahi o nā hiʻohiʻona maikaʻi loa o ka hoʻohana ʻana i ka trigonometry i ke kuhikuhipuʻuone ʻo ia nā Pyramids o ʻAigupita. Ma kēia ʻatikala, e kūkākūkā mākou i nā lakio trigonometric me ka hoʻohana ʻana i nā hiʻohiʻona e pili ana i nā pyramids.
Hoʻolauna i ka Trigonometry i loko o nā Pyramids
ʻO nā pyramid ʻAigupita, ʻoi aku hoʻi nā Pyramids o Giza, he mau hale kaulana loa ia a ua lilo i kumuhana o ke aʻo ʻana e nā makemakika a me nā mea kālai hale he nui. ʻO kekahi o nā ʻāpana koʻikoʻi o kahi pyramid ʻo ia ka huinakolu. Hiki ke loaʻa nā huinakolu ma nā hiʻohiʻona profile a me nā ʻāpana kea.
Mai kahi pyramid, hiki iā mākou ke loaʻa nā huinakolu ʻākau, nā huinakolu equilateral, a me nā ʻano huinakolu like ʻole. He mea kōkua nui ka hoʻohana ʻana o ka trigonometry i ka hoʻoholo ʻana i nā ana, ke kiʻekiʻe, a me ka pali o kahi pyramid.
Laʻana pilikia
Nīnau 1: Ke helu ʻana i ke kiʻekiʻe o kahi Pyramid
"Manaʻo ʻia he 150 mika ka lōʻihi o ke kumu o kahi pyramid a he 130 mika ka lihi o ka ʻaoʻao (apothem). He aha ke kiʻekiʻe o ka pyramid?"
Kūkākūkā:
Ma kēia pilikia, ua hāʻawi ʻia mai iā mākou ka hypotenuse a me ka lōʻihi o ke kumu o kahi pyramid. No ka helu ʻana i ke kiʻekiʻe o ka pyramid, hiki iā mākou ke hoʻohana i ka Pythagorean theorem. Hiki ke puʻunaue ʻia ka pyramid i ʻelua mau huinakolu ʻākau.
1. Pono kākou e ʻimi i ka hapalua o ka lōʻihi o ka ʻaoʻao kumu e hana ai i kahi huinakolu ʻākau.
\( \text{Hapalua o ka lōʻihi o ka ʻaoʻao kumu} = \frac{150}{2} = 75 \text{ mika} \)
2. Ua ʻike mākou:
\( a^2 + b^2 = c^2 \)
kahi ʻo \(a\) ka hapalua o ka lōʻihi o ka ʻaoʻao kumu, ʻo \(b\) ke kiʻekiʻe o ka pyramid, a ʻo \(c\) ka hypotenuse.
3. E hoʻokomo i nā helu i loko o ka hoohalike:
\( 75^2 + b^2 = 130^2 \)
4. E helu:
\( 5625 + b^2 = 16900 \)
\( b^2 = 16900 – 5625 \)
\( b^2 = 11275 \)
\( b = \sqrt{11275} \approx 106.2 \text{ mika} \)
No laila, ʻo ke kiʻekiʻe o ka pyramid ma kahi o 106.2 mika.
Nīnau 2: Ke helu ʻana i ke kihi o ka ʻāʻī o kahi Pyramid
"He aha ke kihi o ka hilig o ka apothema o kahi pyramid i ke kumu o kahi pyramid nona ka lōʻihi o ka ʻaoʻao kumu he 150 mika a me ke kiʻekiʻe o 106.2 mika?"
Kūkākūkā:
No ka loaʻa ʻana o ke kihi o ka inclination (\(\theta\)) o ka apothem i ke kumu o ka pyramid, hiki iā mākou ke hoʻohana i ka hana trigonometric, ʻo ia hoʻi ka tangent (\(\tan\)).
1. E hoʻohana i ke ʻano hana \(\tan(\theta) = \frac{\text{height}}{\frac{\text{base}}{2}}\).
2. E hoʻokomo i nā helu:
\( \tan(\theta) = \frac{106.2}{75} \)
3. E helu:
\( \tan(\theta) \approx 1.416 \)
4. E huli i ke kihi me ka hoʻohana ʻana i ka inverse tangent (\(\tan^{-1}\)):
\( \theta = \tan^{-1}(1.416) \approx 54.14^\circ \)
No laila, ʻo ke kihi o ka inclination o ka apothem i ke kumu o ka pyramid ma kahi o 54.14 degere.
Nīnau 3: Ke helu ʻana i ka lōʻihi o ka Apothem me ka Sine a me Cosine
"Manaʻo ʻia he 120 mika ke kiʻekiʻe o kahi pyramid a he 55 degere ke kihi o ka piʻi ʻana o ka apothem o ka pyramid i ke kumu. He aha ka lōʻihi o ka apothem?"
Kūkākūkā:
Hiki iā mākou ke hoʻohana i ka hana sine a cosine paha e hoʻoponopono i kēia pilikia.
1. E hoʻohana i ke cosine e hoʻoponopono iā ia, e hoʻomanaʻo ana:
\( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
2. E hoʻonohonoho hou i ka hoʻohālikelike no ka Hypotenuse (apothem):
\( \text{Hypotenuse} = \frac{\text{Adjacent}}{\cos(\theta)} \)
3. E hoʻokomo i nā helu:
\( \text{Hypotenuse} = \frac{120}{\cos(55^\circ)} \)
4. E helu:
\( \cos(55^\circ) \approx 0.5736 \)
\( \text{Hypotenuse} = \frac{120}{0.5736} \approx 209.3 \text{ mika} \)
No laila, ʻo ka lōʻihi o ka apothem ma kahi o 209.3 mika.
Ka hopena
Ma nā pilikia ma luna, ua hoʻopili mākou i nā lakio trigonometric like ʻole e helu i ke kiʻekiʻe, ke kihi pali, a me ka lōʻihi apothem o kahi pyramid. Me ka hoʻomaopopo ʻana i ka trigonometry, hiki iā mākou ke hoʻoponopono i nā pilikia geometric like ʻole e like me ka paʻakikī i ka nānā mua ʻana. Hāʻawi ka Trigonometry i kahi mea hana waiwai nui no ka hoʻomaopopo ʻana a me ka hoʻoponopono ʻana i nā pilikia a mākou e kū nei i ka honua maoli, ʻoi aku hoʻi i nā ʻano kuhikuhipuʻuone e like me nā pyramids ʻAigupita.
