Mafunso ndi Kukambirana za Mtundu Umodzi wa Ziŵerengero za Trigonometric: tan θ
Trigonometry ndi nthambi ya masamu yomwe imaphunzira ubale womwe ulipo pakati pa ma angles ndi kutalika kwa mbali mu ma triangles. Chimodzi mwa ma trigonometry ratio omwe amakambidwa kawirikawiri ndi tangent (tan). M'nkhaniyi, tiyang'ana kwambiri pakugwiritsa ntchito tan ratio m'mavuto osiyanasiyana ndikukambirana zitsanzo zingapo zokhudzana ndi tan θ.
Tanthauzo la tan θ
Tangent ya ngodya θ imafotokozedwa ngati chiŵerengero cha kutalika kwa mbali inayo ndi kutalika kwa mbali yoyandikana nayo mu kansalu kolondola. Mwa masamu, izi zalembedwa motere:
\[ \tan θ = \frac{\text{opposite side}}{\text{adjacent side}} \]
Mu bwalo la unit, tan ingathenso kutanthauziridwa ngati chiŵerengero pakati pa y coordinate (mbali yakutsogolo) ndi x coordinate (mbali yakumbali) ya mfundo pa bwalo lomwe lili kutali ndi pakati.
Ntchito ya utoto mu Masamu ndi Fiziki
Trigonometry, makamaka ntchito ya tan, imagwiritsidwa ntchito m'masamu ndi m'thupi osiyanasiyana. Mwachitsanzo, mu fizikisi yakale, ntchito ya tan imagwiritsidwa ntchito pofufuza kayendedwe ka projectile, ndipo mu uinjiniya, imagwiritsidwa ntchito kuwerengera ngodya ya kupendekera kapena kupendekera kwa pamwamba.
Mafunso ndi Kukambirana Zitsanzo
Nazi zitsanzo za mafunso ndi zokambirana zawo kuti mumvetse bwino kugwiritsa ntchito tan θ.
Funso 1: Kuwerengera tan θ ya kansalu kolondola
Choperekedwa: Katatu kolondola kali ndi kutalika kwa mbali yakutsogolo moyang'anizana ndi θ ya 4 cm ndi kutalika kwa mbali yoyandikana ndi ngodya ya θ ya 3 cm. Werengani mtengo wa tan θ.
Kukambirana:
Gwiritsani ntchito tanthauzo la tan:
\[ \tan θ = \frac{\text{front side}}{\text{side side}} \]
M'malo mwa mfundo zodziwika bwino:
\[ \tan θ = \frac{4}{3} \]
Kotero, mtengo wa tan θ ndi \( \frac{4}{3} \).
Funso 2: Kudziwa kutalika kwa mbali pogwiritsa ntchito tan θ
Kuperekedwa: Katatu kolondola komwe kali ndi ngodya θ kumadziwika kuti tan θ = 0.75. Kutalika kwa mbali yoyandikana ndi ngodya θ ndi 8 cm. Werengani kutalika kwa mbali yotsutsana ndi ngodya θ.
Kukambirana:
Gwiritsani ntchito tanthauzo la tan kuti mupeze kutalika kwa mbali inayo:
\[ \tan θ = \frac{\text{front side}}{\text{side side}} \]
\[ 0.75 = \frac{\text{front side}}{8} \]
Chulukitsani mbali zonse ziwiri ndi 8 kuti muthetse vutoli.
\[ \malemba{mbali yakutsogolo} = 0.75 \nthawi 8 \]
\[ \malemba{mbali yakutsogolo} = 6 cm \]
Kotero, kutalika kwa mbali yakutsogolo ndi 6 cm.
Funso 3: Kuwerengera ngodya θ ngati tan θ imadziwika
Kuperekedwa: Katatu kolondola kamadziwika kuti tan θ = 1. Tchulani ngodya θ.
Kukambirana:
Kuchuluka kwa ngodya kumakhala kofanana ndi 1 pamene mbali inayo ndi mbali yoyandikana nayo zili zofanana kutalika. Mu trigonometry yoyambira, izi zimachitika pa ngodya ya 45°.
Chifukwa chake, mtengo wa θ ndi 45°.
Funso 4: Kugwiritsa ntchito Tan θ mu mavuto a algebra
Kuperekedwa: Chingwe chimamangiriridwa kuchokera pamwamba pa ndodo yomwe ili ndi kutalika kwa mamita 15 mpaka pamalo ena pansi omwe ali mamita 20 kuchokera pansi pa ndodo. Werengani tan θ, komwe θ ndi ngodya yopangidwa ndi chingwe ndi ndodo.
Kukambirana:
Gwiritsani ntchito tanthauzo la tan:
\[ \tan θ = \frac{\text{front side (pole height)}}{\text{side side (horizontal distance)}} \]
\[ \tan θ = \frac{15}{20} \]
Chepetsani gawo:
\[ \tan θ = \frac{3}{4} \]
Kotero, mtengo wa tan θ ndi \( \frac{3}{4} \).
Funso 5: Kudziwa kutalika kuchokera pa mtunda ndi ngodya yopendekera
Kuperekedwa: Woyang'anira ali pamtunda wa mamita 100 kuchokera ku nyumba yayitali. Tani θ ya kuwona kuchokera pamalo a wowonera mpaka pamwamba pa nyumbayo ndi \(\tan 30^\circ\). Dziwani kutalika kwa nyumbayo.
Kukambirana:
Zimadziwika kuti \(\tan 30^\circ = \frac{1}{\sqrt{3}}\).
\[ \tan θ = \frac{\text{front side (building height)}}{\text{side side (distance)} } \]
Ikani mfundo zodziwika bwino mu equation
\[ \frac{1}{\sqrt{3}} = \frac{\text{building height}}{100} \]
Chulukitsani mbali zonse ziwiri ndi 100 kuti mupatule kutalika.
\[ \text{building height} = \frac{100}{\sqrt{3}} \]
\[ \text{building height} = \frac{100 \times \sqrt{3}}{3} \]
\[ \text{building height} ≈ 57.73 \text{meters} \]
Choncho, kutalika kwa nyumbayo ndi pafupifupi mamita 57.73.
Funso 6: Kudziwa ngodya kuchokera kutalika ndi mtunda
Taonani: Mukudziwa kuti kutalika kwa nsanja ndi mamita 50 ndipo mtunda wopingasa kuchokera pamalo owonera mpaka pansi pa nsanja ndi mamita 70. Dziwani ngodya ya kukwera mpaka pamwamba pa nsanja.
Kukambirana:
\[ \tan θ = \frac{\text{tower height}}{\text{horizontal distance}} \]
\[ \tan θ = \frac{50}{70} \]
\[ \tan θ = \frac{5}{7} \]
Kuti tipeze θ, timagwiritsa ntchito ntchito ya inverse tangent (tan⁻¹) kapena arctan.
\[ θ = \tan⁻¹ (\frac{5}{7}) \]
Pogwiritsa ntchito calculator kapena tebulo la trigonometry, titha kupeza mtengo wa θ.
\[θ ≈ 35.54° \]
Kotero, ngodya yokwera pamwamba pa nsanja ndi pafupifupi 35.54°.
Mapeto
Trigonometry ndi chida champhamvu m'magawo ambiri a sayansi. Mwachitsanzo, tangent ndi chiŵerengero chosavuta koma champhamvu chomwe chingagwiritsidwe ntchito kuthetsa mavuto osiyanasiyana okhudzana ndi ma angles ndi kutalika kwa mbali. Mwa kumvetsetsa tanthauzo lake ndi momwe tingagwiritsire ntchito, titha kuthetsa mavuto osiyanasiyana a geometry ndi physics. Mwa kuchita mavuto monga chitsanzo pamwambapa, titha kukhala aluso kwambiri pogwiritsa ntchito tan θ pakuwerengera tsiku ndi tsiku.
