Tambayoyi da Tattaunawa kan Nau'i ɗaya na Ra'ayoyin Trigonometric: tan θ
Trigonometry reshe ne na lissafi wanda ke nazarin alaƙar da ke tsakanin kusurwoyi da tsawon gefe a cikin alwatika. Wani rabon trigonometric da ake yawan magana a kai shine tangent (tan). A cikin wannan labarin, za mu mayar da hankali kan amfani da rabon tan a cikin nau'ikan matsaloli daban-daban kuma mu tattauna misalai da yawa da suka shafi tan θ.
Ma'anar tan θ
An bayyana tangent na kusurwa θ a matsayin rabon tsawon gefen da ke gaba da tsawon gefen da ke maƙwabtaka a cikin alwatika na dama. A lissafi, an rubuta wannan kamar haka:
\[ \tan θ = \frac{\text{gefen gaba}}{\text{gefen kusa}} \]
A cikin da'irar naúrar, ana iya fassara tan a matsayin rabo tsakanin daidaitawar y (gefen gaba) da daidaitawar x (gefen gefe) na wani wuri a kan da'irar da ke da raka'a ɗaya daga tsakiya.
Aikin tan a fannin lissafi da kimiyyar lissafi
Ana amfani da Trigonometry, musamman aikin tan, a fannoni daban-daban na lissafi da na zahiri. Misali, a kimiyyar lissafi ta gargajiya, ana amfani da aikin tan wajen nazarin motsin harsashi, kuma a fannin injiniyanci, ana amfani da shi don ƙididdige kusurwar karkata ko gradient na saman.
Tambayoyi da Tattaunawa Samfura
Ga wasu misalai na tambayoyi da tattaunawarsu don fahimtar amfani da tan θ sosai.
Tambaya ta 1: Lissafin tan θ na alwatika mai kusurwa uku na dama
An Bada: Alwatika mai kusurwa ta dama yana da tsawon gefen gaba na kusurwar da ke gabansa θ na 4 cm da kuma tsawon gefen da ke kusa da kusurwar θ na 3 cm. Lissafa darajar tan θ.
Tattaunawa:
Yi amfani da ma'anar launin ruwan kasa:
\[ \tan θ = \frac{\text{gefen gaba}}{\text{gefen gefe}} \]
Maye gurbin dabi'un da aka sani:
\[ \tan θ = \frac{4}{3} \]
Don haka, ƙimar tan θ shine \( \frac{4}{3} \).
Tambaya ta 2: Ƙayyade tsawon gefe ta amfani da launin tan θ
An Bada: An san cewa alwatika mai kusurwa θ yana da launin tan θ = 0.75. Tsawon gefen da ke kusa da kusurwa θ shine 8 cm. Lissafa tsawon gefen da ke gaba da kusurwar θ.
Tattaunawa:
Yi amfani da ma'anar launin ruwan kasa don nemo tsawon gefen da ke gaba da juna:
\[ \tan θ = \frac{\text{gefen gaba}}{\text{gefen gefe}} \]
\[ 0.75 = \frac{\text{gefen gaba}}{8} \]
A ninka bangarorin biyu da 8 domin a warware lissafin.
\[ \text{gefen gaba} = 0.75 \sau 8 \]
\[ \text{gefen gaba} = 6 cm \]
Don haka, tsawon gefen gaba shine 6 cm.
Tambaya ta 3: Lissafin kusurwa θ idan an san launin tan θ
An bayar: An san cewa alwatika mai kusurwar dama yana nuna launin tan θ = 1. Ka ambaci kusurwar θ.
Tattaunawa:
Tan na kusurwa daidai yake da 1 lokacin da gefen da ke gaba da gefen da ke kusa suka yi daidai a tsayi. A cikin trigonometry na asali, wannan yana faruwa a kusurwar 45°.
Saboda haka, ƙimar θ shine 45°.
Tambaya ta 4: Amfani da Tan θ a cikin matsalolin algebra
An Bada: Ana ɗaure igiya daga saman sandar da ke da tsayin mita 15 zuwa wani wuri a ƙasa wanda yake da nisan mita 20 daga tushen sandar. Lissafa tan θ, inda θ shine kusurwar da igiya da sandar suka samar.
Tattaunawa:
Yi amfani da ma'anar launin ruwan kasa:
\[ \tan θ = \frac{\text{gefen gaba (tsawon sanda)}}{\text{gefen gefe (nisa a kwance)}} \]
\[ \tan θ = \frac{15}{20} \]
Sauƙaƙa rabon:
\[ \tan θ = \frac{3}{4} \]
Don haka, ƙimar tan θ shine \( \frac{3}{4} \).
Tambaya ta 5: Ƙayyade tsayi daga nisan da kusurwar karkata
An bayar: Mai lura yana tsaye mita 100 daga wani dogon gini. Tan θ na abin lura daga matsayin mai lura zuwa saman ginin shine \(\tan 30^\cir\). A tantance tsayin ginin.
Tattaunawa:
An san cewa \(\tan 30^\circ = \frac{1}{\sqrt{3}}\).
\[ \tan θ = \frac{\text{gefen gaba (tsawon gini)}}{\text{gefen gefe (nisa)} } \]
Sanya ƙimar da aka sani a cikin lissafi
\[ \frac{1}{\sqrt{3}} = \frac{\text{tsawon gini}}{100} \]
A ninka bangarorin biyu da 100 domin a ware tsayin.
\[ \text{tsawon gini} = \frac{100}{\sqrt{3}} \]
\[ \text{tsawon gini} = \frac{100 \times \sqrt{3}}{3} \]
\[ \text{tsawon gini} ≈ 57.73 \text{ mita} \]
Don haka, tsayin ginin yana da kimanin mita 57.73.
Tambaya ta 6: Tantance kusurwar daga tsayi da nisa
An ba da: Ka san tsayin hasumiya mita 50 ne kuma nisan kwance daga wurin lura zuwa ƙasan hasumiya mita 70 ne. Ka ƙayyade kusurwar tsayin zuwa saman hasumiyar.
Tattaunawa:
\[ \tan θ = \frac{\text{tsawon hasumiya}}{\text{nisa a kwance}} \]
\[ \tan θ = \frac{50}{70} \]
\[ \tan θ = \frac{5}{7} \]
Domin nemo θ, muna amfani da aikin tangent mai juyawa (tan⁻¹) ko arctan.
\[ θ = \tan⁻¹ (\frac{5}{7}) \]
Ta amfani da kalkuleta ko teburin trigonometry, za mu iya samun ƙimar θ.
\[ θ ≈ 35.54° \]
Don haka, kusurwar ɗagawa zuwa saman hasumiyar tana da kusan 35.54°.
Kammalawa
Trigonometry kayan aiki ne mai ƙarfi a fannoni da yawa na kimiyya. Misali, tangent wani rabo ne mai sauƙi amma mai ƙarfi wanda za a iya amfani da shi don magance matsaloli daban-daban da suka shafi kusurwoyi da tsayin gefe. Ta hanyar fahimtar ma'anarsa da yadda ake amfani da shi, za mu iya magance matsaloli iri-iri na lissafi da kimiyyar lissafi. Ta hanyar yin aiki da matsaloli kamar misalin da ke sama, za mu iya ƙara ƙwarewa wajen amfani da tan θ a cikin lissafin yau da kullun.
