Nā Kiʻi o nā Hana Trigonometric
ʻO nā hana trigonometric, i loaʻa mai ka pilina ma waena o nā kihi a me nā ʻaoʻao o nā huinakolu, he mea nui ia i ka makemakika, ka ʻepekema, a me ka ʻenekinia. ʻO kēia mau hana—sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), a me cotangent (cot)—he mau manawa, ʻo ia hoʻi ke hana hou nei lākou i kā lākou mau waiwai i nā manawa maʻamau. He mea nui ka hoʻomaopopo ʻana i kā lākou mau kiʻi no nā noi he nui, mai ka hana ʻana i nā hōʻailona a hiki i ke aʻo ʻana i nā hanana manawa ma ke ʻano.
1. ʻO ka Pōʻai Unit a me nā Hana Trigonometric
Hiki ke ʻike ʻia nā hana trigonometric me ka hoʻohana ʻana i ka pōʻai unit—he pōʻai o ka radius i kikowaena ʻia ma ke kumu o kahi mokulele hoʻonohonoho. ʻO kahi kihi θ, i ana ʻia ma nā radians, ua kahili ʻia mai ka axis x maikaʻi. ʻO nā hoʻonohonoho (x, y) o ke kiko kahi e hui ai ka ʻaoʻao hope o ke kihi i ka pōʻai unit e hāʻawi i ka ʻike waiwai:
– x = cos(θ)
– y = sin(θ)
2. Ke kaha kiʻi ʻana i nā hana Sine a me Cosine
ʻO nā hana sine a me cosine, he mea nui i ka trigonometry, ua hoʻokaʻawale ʻia e ko lākou mau ʻano like me ka nalu.
Ka Hana Sine
Hoʻomaka ka pakuhi o y = sin(θ) ma ke kumu (0,0) a hahai i kahi nalu mālie a hoʻomau:
– Ka Wā: ʻO ka manawa no hoʻokahi pōʻaiapuni piha o ka hana sine he 2π. ʻO ia hoʻi, hana hou ʻo y = sin(θ) i kēlā me kēia 2π radians.
– Amplitude: ʻO nā waiwai kiʻekiʻe a me ka liʻiliʻi o sin(θ) he 1 a me -1, kēlā me kēia.
– Nā Manaʻo Koʻikoʻi: ʻO nā manaʻo koʻikoʻi ma ka pakuhi e komo pū me (0,0), (π/2,1), (π,0), (3π/2,-1), a me (2π,0).
Ma ke ʻano kiʻi, holo mālie ka nalu sine ma waena o -1 a me 1, e hele ana ma ka axis-x ma nā ʻano he nui o π a hiki i kona kiʻekiʻe a me ka liʻiliʻi loa ma nā ʻano he nui ʻē o π/2.
Ka Hana Cosine
Ua like ke ʻano o ka hana cosine y = cos(θ) me ka hana sine akā ua neʻe ʻia ma ka ʻaoʻao:
– Ka Wā: E like me ka hana sine, he 2π ka wā o ka hana cosine.
– Amplitude: ʻO ka laulā o nā waiwai no cos(θ) mai -1 a i 1 hoʻi.
– Nā Manaʻo Koʻikoʻi: ʻO nā manaʻo koʻikoʻi ma ka pakuhi e komo pū me (0,1), (π/2,0), (π,-1), (3π/2,0), a me (2π,1).
Ma ke ʻano o ke ʻano, hoʻomaka ka nalu cosine ma kona waiwai kiʻekiʻe loa o 1. Hōʻike ke ʻano nalu i ka periodicity like a me ka amplitude e like me ka hana sine akā ua hoʻoneʻe ʻia i ka hema e π/2 radians.
3. Ke kaha kiʻi ʻana i nā hana Tangent a me Cotangent
Ka Hana Tangent
Hōʻike ka pakuhi o y = tan(θ) i ka lakio o ka sine i ke cosine:
– Kau: ʻO ka wā no hoʻokahi pōʻaiapuni piha o ka hana tangent ʻo π.
– Nā Asymptotes: ʻAʻole i wehewehe ʻia ka tangent kahi cos(θ) = 0, e hopena ana i nā asymptotes vertical ma θ = (n+1/2)π kahi ʻo n kekahi helu holoʻokoʻa.
– Nā Manaʻo Koʻikoʻi: ʻO nā manaʻo koʻikoʻi e komo pū me (0,0), (π/4,1), a me (-π/4,-1).
Hele ka pakuhi tangent ma o ke kumu a hoʻonui me ke kaupalena ʻole ʻia i kona hoʻokokoke ʻana i nā asymptotes, e hana ana i kahi moʻo o nā piʻo e hana hou ʻia e like me kahi pūʻulu o nā hyperbolas.
Ka Hana Cotangent
Hōʻike ka hana cotangent y = cot(θ) = cos(θ)/sin(θ) i kekahi ʻano ʻokoʻa:
– Ka wā: ʻO ka wā o ke cotangent ʻo π.
– Nā Asymptotes: ʻAʻole i wehewehe ʻia ʻo Cotangent kahi sin(θ) = 0, e alakaʻi ana i nā asymptotes kū pololei ma nā multiples o π.
– Nā Manaʻo Koʻikoʻi: ʻO nā ʻano e komo pū me θ = (π/4,1), (3π/4,-1), a pēlā aku.
Ma ke kaha kiʻi ʻana i ka cotangent, ʻike mākou i nā piʻo hoʻomau e like me nā piʻo tangent akā ua hoʻohuli ʻia a hōʻike ʻia ma muli o ke ʻano o nā hana reciprocal.
4. Ke kaha kiʻi ʻana i nā hana Cosecant a me Secant
ʻO ka Hana Cosecant
ʻO ka hana y = csc(θ) = 1/sin(θ) e lawe mai i nā mea like:
– Nā Asymptotes: Loaʻa i ka hana cosecant nā asymptotes kū pololei kahi sin(θ) = 0, e kū ana ma nā multiples o π.
– ʻAno: ʻOiai ʻo y = csc(θ) ke kūkaʻi o y = sin(θ), i ka wā e liʻiliʻi loa ai ka sin(θ), lilo ka csc(θ) i mea nui loa, a ʻo ka hope.
Hōʻike ʻia ka pakuhi cosecant e kahi moʻo o nā piʻo e hoʻokokoke aku ana i ka palena ʻole kokoke i nā asymptotes, e hana ana i kahi ʻano o nā ʻano parabolic i ulana ʻia ma waena o nā asymptotes.
Ka Hana Secant
Pēlā nō, ʻike ʻia ʻo y = sec(θ) = 1/cos(θ) ma ke ʻano he reciprocal o ka hana cosine:
– Nā Asymptotes: Loaʻa kēia mau mea ma nā wahi kahi cos(θ) = 0, ʻo ia hoʻi, kahi θ = (2n+1)π/2.
– ʻAno: Hōʻike ka pakuhi i nā waiwai nui kokoke i nā asymptotes kū pololei a me nā waiwai liʻiliʻi ma waena o lākou.
ʻO ka pakuhi hopena o ka secant he mau lālā parabolic e kū pono ana i ka pakuhi cosine, e hōʻike ana i nā pilina paʻakikī ma waena o nā hana trigonometric reciprocal a me kā lākou mau hana makua.
5. Hoololi
Hiki ke hoʻololi ʻia, hoʻolōʻihi ʻia, a hōʻike ʻia nā hana trigonometric ma ka hoʻololi ʻana i kā lākou mau kaulike kumu:
– Nā Hoʻololi Kū pololei: ʻO ka hoʻohui ʻana i kahi kūpaʻa, e like me y = sin(θ) + c, e hoʻoneʻe i ka pakuhi i luna a i lalo paha e nā anakahi c.
– Nā Hoʻololi Paepae: ʻO ka hoʻohui/hoʻemi ʻana i loko o ka hana, e like me y = sin(θ – d), hoʻoneʻe i ka pakuhi i ka hema a i ʻole ka ʻākau.
– Amplitude: Ke hoʻonui nei i ka hana, e laʻa, y = a sin(θ), hoʻololi i ke kiʻekiʻe o nā piko a me nā awāwa.
– Nā Hoʻoponopono Kau: ʻO ka hoʻololi ʻana i ke alapine (frequency), e laʻa, y = sin(bθ), e hoʻopilikia i ka helu o nā pōʻaiapuni i loko o kahi manawa i hāʻawi ʻia. No ka laʻana, ʻo y = sin(2θ) e hana i kahi nalu me ka hapalua o ka wā o ka nalu sine mua.
Panina
Hāʻawi nā kiʻi o nā hana trigonometric i nā mea hana ʻike maka ikaika no ka hoʻomaopopo ʻana i nā hanana periodic. ʻO ka ʻike pono ʻana i kēia mau kiʻi e pili ana i ka ʻike ʻana i nā ʻano kumu o ka sine, cosine, tangent, a me kā lākou reciprocals, me ka hoʻomaopopo ʻana i nā hoʻololi e hoʻoponopono i ko lākou ʻano a me ke kūlana. Me nā noi e uhi ana i nā ʻano aʻo like ʻole, ʻo ka hoʻomaopopo piha ʻana i kēia mau hōʻike kiʻi e waiho ana i kahi kahua koʻikoʻi no ke aʻo ʻana i nā pae kiʻekiʻe a me ka hoʻokō pono ʻana i nā kahua like ʻole e like me ka physics, ka ʻenekinia, a ma waho aku.
