Imibuzo Eyisibonelo Exoxa Ngokusetshenziswa Kwezilinganiso Ze-Trigonometric tan θ
I-Trigonometry iyigatsha lezibalo elibhekene nama-engeli nemisebenzi yama-engeli konxantathu. Umqondo owodwa obalulekile ku-trigonometry yizilinganiso ze-trigonometric zama-engeli, njenge-sine (sin), i-cosine (cos), kanye ne-tangent (tan). Kulesi sihloko, sizogxila ku-tangent ye-engeli eyodwa θ, ekhonjiswa yi-tan θ.
Incazelo ye-Tan θ
I-tangent ye-engeli θ kunxantathu ongakwesokudla iyisilinganiso sobude bohlangothi oluphambene lwe-engeli θ nobude bohlangothi oluseduze lwe-engeli θ. Ngokwezibalo, i-tan θ ivezwa kanje:
\[ \tan \theta = \frac{\text{uhlangothi oluphambene lwe-engeli θ}}{\text{uhlangothi oluseduze lwe-engeli θ}} \]
Ukuze siqonde kangcono lo mqondo, sizodlula ezinkingeni ezithile zesibonelo bese sixoxa ngokusetshenziswa kwe-tan θ.
Isibonelo Umbuzo 1: Ukubala u-Tan θ
Uma unikezwe unxantathu ongakwesokudla one-engeli engu-θ endaweni ethi A, lapho uhlangothi oluphambene lwe-engeli engu-θ lunobude obungu-3 cm kanti uhlangothi oluseduze lwe-engeli engu-θ lunobude obungu-4 cm. Bala i-tan θ.
Isixazululo:
Kusukela ezinkingeni ezingenhla, siyazi:
– Uhlangothi oluphambene lwe-engeli θ (oluphambene) = 3 cm
– Uhlangothi oluseduze lwe-engeli θ = 4 cm
Sisebenzisa incazelo ye-tan θ, sibala:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
\[ \tan \theta = \frac{3}{4} \]
Ngakho-ke, i-tan θ = 0.75.
Ngokwejiyometri, lokhu kusho ukuthi nge-engeli engu-θ kunxantathu, isilinganiso sobude bohlangothi oluphambene nobude bohlangothi oluseduze singu-0.75.
Isibonelo 2: Ukusebenzisa i-Tan θ ukuze ubale ubude obuseceleni
Iladi lincike odongeni nge-engeli yokuphakama engu-θ engama-degree angu-30. Ibanga ukusuka onyaweni lweladi kuya odongeni lingamamitha angu-5. Lide kangakanani iladi lincike odongeni?
Isixazululo:
Isinyathelo sokuqala, sikhumbula incazelo ye-tan θ:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
Uma kubhekwa le nkinga:
– θ = amadigri angu-30
– eduze (ibanga ukusuka onyaweni lweladi kuya odongeni) = amamitha angu-5
– okuphambene (ukuphakama kwesitebhisi kuya odongeni) = ???
Okokuqala sibala\text{opposite)):
\[ \tan 30^\circ = \frac{\text{opposite}}{5} \]
Siyazi kusukela kuthebula le-trigonometric ukuthi:
\[ \tan 30^\circ = \frac{\sqrt{3}}{3} \]
Ngakho-ke:
\[ \frac{\sqrt{3}}{3} = \frac{\text{opposite}}{5} \]
Phindaphinda izinhlangothi zombili ngo-5:
\[ \text{opposite} = 5 \cdot \frac{\sqrt{3}}{3} = \frac{5\sqrt{3}}{3} \]
Okuphambene (ukuphakama kwesitebhisi odongeni) yilokhu:
\[ \frac{5\sqrt{3}}{3} \cishe 2.89 \text{meters} \]
Ngakho-ke, ubude belethi bungamamitha ama-5.
Isibonelo 3: Ukubala ama-Angles Usebenzisa i-Tan θ
Umbhoshongo udala isithunzi esingamamitha ayi-12 ubude. Uma umbhoshongo ungamamitha ayi-8 ukuphakama, iyini i-engeli yokuphakama kwelanga engu-θ?
Isixazululo:
Kule nkinga, sinikezwa:
– Ukuphakama kombhoshongo (ngokuphambene) = amamitha angu-8
– Ubude besithunzi (eduze) = amamitha ayi-12
Sisebenzisa incazelo ye-tan θ ukuthola i-θ:
\[ \tan \theta = \frac{8}{12} = \frac{2}{3} \]
Manje sithola u-θ nge-equation:
\[ \theta = \tan^{-1} \left(\frac{2}{3}\right) \]
Uma sibheka etafuleni noma emshinini wokubala ukuze sithole inani le-tangent ephambene, sithola:
\[ \theta \cishe 33.69^\circ \]
Ngakho-ke, i-engeli yokuphakama kwelanga icishe ibe ngama-degree angu-33.69.
Isibonelo 4: Ukusebenzisa i-Tan θ Ezidingweni Zezwe Langempela
Kufakwe isibuko sokukhanya esifakwe esigxotsheni esingamamitha angu-4 ngaphezu kwemoto. Uma ufuna ukufaka isibuko esingabonakala nge-engeli engama-degree angu-45 ukusuka phansi, bala ibanga elikhulu kakhulu lapho isibuko singabonakala khona.
Isixazululo:
Kusukela embuzweni, kuyaziwa:
– Ukuphakama kwensika (ngokuphambene) = amamitha angu-4
– I-engela θ = amadigri angu-45
Ngokusho kwencazelo ye-tan θ:
\[ \tan 45^\circ = \frac{\text{opposite}}{\text{adjacent}} \]
Siyazi ukuthi \(\tan 45^\circ = 1\), ngakho-ke:
\[ 1 = \frac{4}{\text{adjacent}} \]
Ngakho-ke:
\[ \umbhalo{oseduze} = 4 \umbhalo{amamitha} \]
Ngakho-ke, ibanga elide kakhulu lapho i-siren ingabonakala khona ngamamitha angu-4.
Isiphetho
Kusukela ezibonelweni ezingenhla, sibona ukuthi i-tangent ye-angle θ (\(\tan \theta\)) ingumqondo owusizo kakhulu futhi inezinhlobo eziningi zezicelo ezisebenzayo, kusukela ekuxazululeni izinkinga ezilula kwizibalo kuya ekusetshenzisweni kwayo ezidingweni zansuku zonke, njengokwakha kanye nokuzulazula. Ukuqonda kahle lo mqondo kungasiza ekuxazululeni izinkinga ezahlukahlukene ezihilela ukuqhathanisa ubude bezinhlangothi kunxantathu.
Sekukonke, i-tan θ, njengengxenye ye-trigonometry, ayiyona nje indaba ebalulekile emfundweni esemthethweni kodwa futhi iyithuluzi eliwusizo kakhulu ezicini ezahlukene zempilo yangempela. Ngethemba ukuthi lesi sihloko sinikeza umbono ocacile nojulile wendlela yokusebenzisa i-tan θ ukuxazulula izinkinga ezihlobene.
