Zitsanzo za Mafunso Okhudza Kugwiritsa Ntchito Ma Trigonometric Ratios tan θ
Trigonometry ndi nthambi ya masamu yomwe imayang'ana ma angles ndi ntchito za ngodya mu ma triangles. Lingaliro limodzi lofunika mu trigonometry ndi ma ratios a trigonometric a ma angles, monga sine (sin), cosine (cos), ndi tangent (tan). M'nkhaniyi, tikambirana za tangent ya ngodya imodzi θ, yomwe imawonetsedwa ndi tan θ.
Tanthauzo la Tan θ
Kuzungulira kwa ngodya θ mu kansalu kolondola ndi chiŵerengero cha kutalika kwa mbali yotsutsana ya ngodya θ ndi kutalika kwa mbali yoyandikana nayo ya ngodya θ. Mwa masamu, tan θ imafotokozedwa motere:
\[ \tan \theta = \frac{\text{mbali yotsutsana ya ngodya θ}}{\text{mbali yoyandikana ya ngodya θ}} \]
Kuti timvetse bwino lingaliro ili, tikambirana za mavuto ena ndi kukambirana za momwe tan θ imagwirira ntchito.
Chitsanzo Funso 1: Kuwerengera Tan θ
Popeza pali kansalu kakumanja komwe kali ndi ngodya ya θ pamalo A, pomwe mbali inayo ya ngodya ya θ ili ndi kutalika kwa masentimita 3 ndipo mbali yoyandikana nayo ya ngodya ya θ ili ndi kutalika kwa masentimita 4. Werengani tan θ.
Yankho:
Kuchokera pa mavuto omwe ali pamwambapa, tikudziwa:
– Mbali yotsutsana ya ngodya θ (yotsutsana) = 3 cm
– Mbali yapafupi ya ngodya θ = 4 cm
Pogwiritsa ntchito tanthauzo la tan θ, timawerengera:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
\[ \tan \theta = \frac{3}{4} \]
Kotero, tan θ = 0.75.
Mwa geometriki, izi zikutanthauza kuti pa ngodya θ mu katatu, chiŵerengero cha kutalika kwa mbali inayo ndi kutalika kwa mbali yapafupi ndi 0.75.
Chitsanzo 2: Kugwiritsa ntchito Tan θ powerengera kutalika kwa mbali
Makwerero akutsamira khoma pa ngodya yokwera ya madigiri θ ya madigiri 30. Mtunda wochokera pansi pa makwerero kupita ku khoma ndi mamita 5. Kodi makwerero akutsamira khoma ndi aatali bwanji?
Yankho:
Gawo loyamba, tikukumbukira tanthauzo la tan θ:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
Pankhani ya vutoli:
– θ = madigiri 30
– pafupi (mtunda kuchokera pansi pa makwerero kupita kukhoma) = mamita 5
– moyang'anizana (kutalika kwa makwerero kupita kukhoma) = ???
Choyamba timawerengera\text{opposite)):
\[ \tan 30^\circ = \frac{\text{opposite}}{5} \]
Tikudziwa kuchokera pa tebulo la trigonometric kuti:
\[ \tan 30^\circ = \frac{\sqrt{3}}{3} \]
Kotero:
\[ \frac{\sqrt{3}}{3} = \frac{\text{opposite}}{5} \]
Chulukitsani mbali zonse ziwiri ndi 5:
\[ \text{opposite} = 5 \cdot \frac{\sqrt{3}}{3} = \frac{5\sqrt{3}}{3} \]
Mosiyana (kutalika kwa makwerero kupita kukhoma) ndi:
\[ \frac{5\sqrt{3}}{3} \pafupifupi 2.89 \text{meters} \]
Kotero, kutalika kwa makwerero ndi mamita 5.
Chitsanzo 3: Kuwerengera Ma Angles Pogwiritsa Ntchito Tan θ
Nsanja imapanga mthunzi wautali mamita 12. Ngati nsanjayo ili ndi kutalika kwa mamita 8, kodi ngodya ya kukwera kwa dzuwa ndi yotani?
Yankho:
Mu funso ili, tapatsidwa:
– Kutalika kwa nsanja (motsutsana) = mamita 8
– Kutalika kwa mthunzi (pafupi) = mamita 12
Timagwiritsa ntchito tanthauzo la tan θ kuti tipeze θ:
\[ \tan \theta = \frac{8}{12} = \frac{2}{3} \]
Tsopano tikupeza θ ndi equation:
\[ \theta = \tan^{-1} \left(\frac{2}{3}\right) \]
Tikayang'ana tebulo kapena chowerengera kuti tidziwe mtengo wa inverse tangent, timapeza:
\[ \theta \pafupifupi 33.69^\circ \]
Kotero, ngodya ya kukwera kwa dzuwa ndi pafupifupi madigiri 33.69.
Chitsanzo 4: Kugwiritsa Ntchito Tan θ pa Zosowa Zenizeni za Dziko
Chowunikira chowala chomwe chili pa ndodo ya mamita 4 pamwamba pa galimoto chayikidwa. Ngati mukufuna kuyika siren yomwe imawoneka pa ngodya ya madigiri 45 kuchokera pansi, werengani mtunda wautali kwambiri womwe siren ingawonekerebe.
Yankho:
Kuchokera pa funsoli, zimadziwika kuti:
– Kutalika kwa nsonga (motsutsana) = mamita 4
– Ngodya θ = madigiri 45
Malinga ndi tanthauzo la tan θ:
\[ \tan 45^\circ = \frac{\text{opposite}}{\text{adjacent}} \]
Tikudziwa kuti \(\tan 45^\circ = 1\), kotero:
\[ 1 = \frac{4}{\text{adjacent}} \]
Kotero:
\[ \malemba{adjacent} = 4 \malemba{mita} \]
Kotero, mtunda wakutali kwambiri womwe siren ingawonekere ndi mamita 4.
Mapeto
Kuchokera ku zitsanzo zomwe zili pamwambapa, tikuwona kuti tangent ya ngodya θ (\(\tan \theta\)) ndi lingaliro lothandiza kwambiri ndipo lili ndi ntchito zosiyanasiyana, kuyambira kuthetsa mavuto osavuta mu masamu mpaka kugwiritsidwa ntchito kwake pa zosowa za tsiku ndi tsiku, monga pakupanga ndi kuyenda. Kumvetsetsa bwino lingaliro ili kungathandize kuthetsa mavuto osiyanasiyana okhudzana ndi kufananiza kutalika kwa mbali mu kansalu kakang'ono.
Ponseponse, tan θ, monga gawo la trigonometry, si nkhani yofunika kwambiri pa maphunziro apamwamba komanso chida chothandiza kwambiri m'mbali zosiyanasiyana za moyo weniweni. Tikukhulupirira kuti nkhaniyi ikupereka chithunzithunzi chomveka bwino komanso chakuya cha momwe mungagwiritsire ntchito tan θ kuthetsa mavuto okhudzana ndi izi.
