Uhlobo Olulodwa Lwezilinganiso Ze-Trigonometric: tan θ
I-Trigonometry iyigatsha lezibalo elifunda ubudlelwano phakathi kwezinhlangothi nama-engeli kanxantathu. Esinye sezilinganiso ze-trigonometric eziyisisekelo nezibaluleke kakhulu yi-tangent, efanekiselwa yi-tan θ. Kulesi sihloko, sizohlola umqondo oyisisekelo we-tangent, indlela yokuyibala, kanye nokusetshenziswa kwayo emikhakheni eyahlukahlukene.
Incazelo ye-Tangent (tan θ)
Ku-trigonometry, i-tangent ye-engeli engu-θ kunxantathu ongakwesokudla ichazwa njengesilinganiso sobude bohlangothi oluphambene ngqo ne-engeli (uhlangothi oluphambene) nobude bohlangothi oluseduze ne-engeli (uhlangothi oluseduze). Ifomula ejwayelekile ithi:
\[ \text{tan } θ = \frac{\text{front side}}{\text{side side}} \]
Isibonelo, kunxantathu ongakwesokudla one-engeli engu-θ, uma uhlangothi oluphambene lunobude u-a kanti uhlangothi oluseduze lunobude u-b, khona-ke:
\[ \umbhalo{tan } θ = \frac{a}{b} \]
Ngaphezu kwalokho, i-tangent ingafanekiswa nangokuqhathanisa i-sine ne-cosine:
\[ \umbhalo{tan } θ = \frac{\umbhalo{sin } θ}{\umbhalo{cos } θ} \]
Ukubala i-Tangent (tan θ)
Ukuze sibale i-tan θ, sidinga ukwazi ubude bezinhlangothi ezimbili ezifanele kunxantathu kanye ne-engeli elinganiswayo. Okokuqala, sidinga ukuqinisekisa ukuthi i-engeli elinganiswayo iyi-engeli kunxantathu ongakwesokudla.
Isibonelo Sokubala
Ake sithi sinonxantathu one-engeli eyodwa θ ngqo maqondana nohlangothi lobude 5 kanye nohlangothi lobude 12. Ukuthola inani le-tan θ:
\[ \umbhalo{tan } θ = \frac{5}{12} \]
Nokho, inani le-tan θ le-engeli θ lingu-5/12 noma u-0.4167.
Uma sinonxantathu lapho ubude bohlangothi oluphambene bungu-3 kanti ubude bohlangothi oluseduze bungu-4, khona-ke:
\[ \umbhalo{tan } θ = \frac{3}{4} = 0.75 \]
Umbono weJiyomethri we-Tangent
Uma sidweba i-tangent ku-diagram ye-trigonometric ngaphakathi kwendilinga yeyunithi, sithola isithombe esiqondakala kalula. Endilinga yeyunithi, i-angle θ ivezwa ngama-radians, kanti i-tangent yaleyo angle ubude bomugqa odwetshwe kusukela ekuqaleni (0,0) kuya ephuzwini (1,tan θ) othinta indilinga.
Umsebenzi we-Inverse Tangent
Ngokusebenza kahle, i-tangent ine-inverse ebizwa ngokuthi i-arctan noma i-atan. Lo msebenzi we-inverse usetshenziselwa ukuthola i-engeli θ uma i-tangent yaleyo engeli iyaziwa. Isisho esijwayelekile sithi:
\[ θ = \text{tan}^{-1}(x) \text{ or } \text{atan}(x) \]
Isibonelo Sokubala
Uma sinenani eliqondile, isibonelo 1, ukuthola i-engeli θ egcwalisa i-tan θ = 1, sisebenzisa umsebenzi ophambene:
\[ θ = \text{tan}^{-1}(1) = 45° \text{ or } \frac{\pi}{4} \text{ radians} \]
Ukusetshenziswa kwe-tangent
Ukusetshenziswa kwe-tangent kusabalala emikhakheni eminingi, kusukela ku-geometry kuya ku-physics, ubunjiniyela, izinkanyezi, ngisho nasemikhakheni efana nezomnotho kanye nezokwelapha.
I-Geodesy kanye neMephu
Okunye ukusetshenziswa kwe-tangent kuseku-geodesy kanye ne-mapping. I-tangent isetshenziselwa ukuthola ukuphakama kwezinto okunzima ukuzilinganisa ngqo. Isibonelo, ukuze kutholakale ukuphakama kombhoshongo, umuntu angalinganisa ibanga eliqondile ukusuka phansi kombhoshongo kuya endaweni yokubuka kanye ne-engeli yokuphakama kusukela endaweni yokubuka kuya phezulu kombhoshongo. Ukuphakama kombhoshongo (H) kungabalwa kanje:
\[ H = D \izikhathi \umbhalo{tan } θ \]
Lapho u-D eyibanga elivundlile kanye no-θ kuyi-elevation angle.
Ifiziksi
Ku-physics, ama-tangent asetshenziswa ekubaleni okuhlukahlukene okubandakanya ama-engeli, ijubane, amandla, kanye nomfutho. Isibonelo, ekuhlaziyweni kokunyakaza kwe-projectile, lapho i-engeli yokuqalisa kanye nejubane lokuqala kuthinta ibanga elihanjiwe.
astronomy
Ama-Tangents asetshenziswa futhi ku-astronomy, ikakhulukazi ekubaleni amabanga ezinkanyezi. Isibonelo, i-parallax yenkanyezi iyi-engeli encane esetshenziswa izazi zezinkanyezi ukukala ibanga lenkanyezi ukusuka eMhlabeni.
Ukuqonda Imiqondo Ngamagrafu
Igrafu yomsebenzi we-tangent inikeza umbono ocacile wokuthi i-tan ishintsha kanjani nge-engeli. Umsebenzi we-tangent une-period \( π \) futhi unezimpawu eziqondile kuzo zonke \( \frac{π}{2} + kπ \), lapho u-k eyinombolo ephelele. Lokhu kubonisa ukuthi i-tan θ ayichazwanga kulezi zi-engeli (ama-engeli angajwayelekile kune-π/2).
Isiphetho
I-tangent ingenye yezilinganiso ze-trigonometric eziyisisekelo neziwusizo. Ukwazi i-tangent ye-engeli kusinika ukuqonda isilinganiso phakathi kwezinhlangothi zonxantathu ongakwesokudla. I-tangent isetshenziswa kabanzi emikhakheni ehlukahlukene yesayensi kanye nemikhuba yansuku zonke, kusukela kumephu yezwe kanye ne-physics kuya ku-astronomy.
Ngokuqonda okujulile kwe-tan θ kanye nokusetshenziswa kwayo, singathuthukisa izinhlelo zokusebenza ezihlakaniphile nezisebenza kahle emikhakheni ehlukahlukene yesayensi nobuchwepheshe. Njengomqondo oyinhloko ku-trigonometry, i-tangent inikeza isisekelo esiqinile sokuqonda nokusebenzisa izimiso zezibalo empilweni yansuku zonke kanye nemikhakha eyahlukene.
