Nā nīnau hoʻohālike Trigonometry a me ke kūkākūkā ʻana
ʻO ka Trigonometry kahi lālā o ka makemakika e aʻo ana i ka pilina ma waena o nā kihi a me nā lōʻihi o nā ʻaoʻao i loko o nā huinakolu. He mea nui ka hoʻomaopopo ʻana i nā manaʻo kumu o ka trigonometry i nā ʻano noi like ʻole, mai ka physics a me ka ʻenekinia a hiki i ka astronomy a me ka geography. Ma kēia ʻatikala, e wehewehe mākou i kekahi mau pilikia hoʻohālike a me nā wehewehe piha e kōkua i ka hoʻomaopopo ʻana.
Laʻana Nīnau 1: Ke Helu ʻana i nā ʻAoʻao o kahi Huikolu me ka hoʻohana ʻana i ke ʻAno Sine
Nīnau:
Hāʻawi ʻia kahi huinakolu ABC, me ke kihi A = 30°, ke kihi B = 45°, a me ka ʻaoʻao b = 10 kenimika. E helu i ka lōʻihi o ka ʻaoʻao a.
Kūkākūkā:
E hoʻohana i ke kānāwai o nā sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
E hoʻokomo i nā waiwai i ʻike ʻia:
\[ \frac{a}{\sin 30°} = \frac{10}{\sin 45°} \]
Ua ʻike mākou:
\[ \sin 30° = \frac{1}{2} \]
\[ \sin 45° = \frac{\sqrt{2}}{2} \]
I kēia manawa, e pani i kēia mau waiwai i loko o ka hoohalike:
\[ \frac{a}{\frac{1}{2}} = \frac{10}{\frac{\sqrt{2}}{2}} \]
E hoʻomaʻalahi i ka hoʻohālikelike:
\[ 2a = \frac{10 \cdot 2}{\sqrt{2}} \]
\[ 2a = \frac{20}{\sqrt{2}} \]
E hoʻomaopopo i ka denominator:
\[ 2a = \frac{20 \cdot \sqrt{2}}{2} \]
\[ 2a = 10\sqrt{2} \]
\[ a = 5\sqrt{2} \]
No laila, ʻo ka lōʻihi o ka ʻaoʻao a he \( 5\sqrt{2} \) cm.
Laʻana Nīnau 2: Ke Helu ʻana i nā Huina me ka hoʻohana ʻana i ke ʻAno Cosine
Nīnau:
He ʻaoʻao ko ka huinakolu ʻo a = 7 kenimika, b = 10 kenimika, a ʻo c = 5 kenimika. E hoʻoholo i ka nui o ke kihi C.
Kūkākūkā:
E hoʻohana i ke kānāwai o nā cosines:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
E hoʻokomo i nā waiwai i ʻike ʻia:
\[ 5^2 = 7^2 + 10^2 – 2 \cdot 7 \cdot 10 \cdot \cos C \]
E hoʻomaʻalahi i ka hoʻohālikelike:
\[ 25 = 49 + 100 – 140 \ko C \]
\[ 25 = 149 – 140 \ko C \]
E neʻe i ka 149 i ka ʻaoʻao hema:
\[ 25 – 149 = -140 \cos C \]
\[ -124 = -140 \cos C \]
\[ \cos C = \frac{124}{140} \]
\[ \cos C = \frac{62}{70} \]
\[ \cos C = \frac{31}{35} \]
E hoʻohana i ka mīkini helu e ʻike ai i ka \( \cos^{-1} \) (inverse cosine):
\[ C \approx \cos^{-1}\left(\frac{31}{35}\right) \]
\[ C \approx 25.84° \]
No laila, ʻo ka nui o ke kihi C ma kahi o 25.84°.
Laʻana Nīnau 3: Ke Helu ʻana i ke Kiʻekiʻe a me ka ʻĀpana o kahi Huikolu
Nīnau:
He ʻelua ʻaoʻao ko ka huinakolu o ka lōʻihi ʻo a = 6 kenimika a ʻo b = 8 kenimika me ke kihi ma waena o lāua, \(\theta\) = 60°. E helu i ke kiʻekiʻe a me ka ʻāpana o ka huinakolu.
Kūkākūkā:
1. Ke helu ʻana i ka ʻāpana o kahi huinakolu:
E hoʻohana i ke ʻano no ka wahi o kahi triangle:
\[ \text{Area} = \frac{1}{2} ab \sin \theta \]
E hoʻokomo i nā waiwai i ʻike ʻia:
\[ \text{Area} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \sin 60° \]
Ua ʻike mākou:
\[ \sin 60° = \frac{\sqrt{3}}{2} \]
No laila:
\[ \text{Area} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{Area} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{ʻĀpana} = 24 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{ʻĀpana} = 12\sqrt{3} \]
No laila, ʻo ka ʻāpana o ka huinakolu he \( 12\sqrt{3} \) cm².
2. Ke helu ʻana i ke kiʻekiʻe o kahi huinakolu mai ke kumu a:
No ka helu ʻana i ke kiʻekiʻe o kahi huinakolu, e hōʻike i ke kiʻekiʻe ma ke ʻano he h a hoʻohana i ke ʻano hana ʻāpana:
\[ \text{Area} = \frac{1}{2} \times a \times h \]
\[ 12\sqrt{3} = \frac{1}{2} \times 6 \times h \]
\[ 12\sqrt{3} = 3h \]
\[ h = \frac{12\sqrt{3}}{3} \]
\[ h = 4\sqrt{3} \]
No laila, ʻo ke kiʻekiʻe o ka huinakolu he \( 4\sqrt{3} \) kenimika.
Laʻana Nīnau 4: Ke Hoʻoholo ʻana i nā ʻAoʻao o kahi Huikolu ʻĀkau
Nīnau:
I loko o kahi huinakolu ʻākau, me ke kihi \(\theta\) = 30° a ʻo ka ʻaoʻao like o ke kihi \(\theta\) he 5 kenimika, e hoʻoholo i ka lōʻihi o ka hypotenuse.
Kūkākūkā:
E hoʻohana i nā lakio trigonometric no kahi kihi 30° i loko o kahi huinakolu ʻākau:
\[ \sin \theta = \frac{\text{front}}{\text{hypotenuse}} \]
\[ \sin 30° = \frac{obverse}{hypotenuse} = \frac{5}{\text{hypotenuse}} \]
Ua ʻike mākou:
\[ \sin 30° = \frac{1}{2} \]
No laila:
\[ \frac{1}{2} = \frac{5}{\text{hypotenuse}} \]
\[ \kikokikona{kahawaena} = 10 \]
No laila, ʻo ka lōʻihi o ka hypotenuse he 10 kenimika.
Laʻana Nīnau 5: Ke helu ʻana i nā kihi me nā hana Trigonometric
Nīnau:
Inā \( \tan \theta = \frac{3}{4} \), e helu i ka nui o ke kihi \(\theta\).
Kūkākūkā:
Ke hoʻohana nei i ke ʻano tangent inverse:
\[ \theta = \tan^{-1} \left(\frac{3}{4}\right) \]
Me ke kōkua o ka mīkini helu:
\[ \theta \approx 36.87° \]
No laila, ʻo ka nui o ke kihi \(\theta\) ma kahi o 36.87°.
Ka hopena
He manaʻo makemakika ākea ka Trigonometry me ka hoʻohana ākea ma nā ʻano like ʻole. ʻO ka hoʻomaopopo ʻana i ka hoʻohana ʻana i nā kānāwai o nā sines, cosines, a me nā hana kumu ʻē aʻe e hiki ai iā mākou ke hoʻoponopono i nā pilikia like ʻole e pili ana i nā huinakolu a me nā kihi. Ma o nā laʻana ma luna, ke manaʻolana ʻia e loaʻa i ka poʻe heluhelu kahi ʻike hohonu a hiki ke hoʻopili iā ia i nā kūlana like ʻole e pono ai ka hoʻomaopopo ʻana i ka trigonometry.
