Mienzaniso yeMibvunzo Inokurukura Kushandiswa kweTrigonometric Ratios tan θ
Trigonometry ibazi remasvomhu rinobata nemakona nemabasa emakona mumatriangles. Imwe pfungwa yakakosha mutrigonometry ndeye trigonometric ratios yemaangles, akadai se sine (sin), cosine (cos), uye tangent (tan). Muchinyorwa chino, tichatarisa pane tangent yekona imwe chete θ, iyo inoratidzwa ne tan θ.
Tsanangudzo yeTan θ
Tangent yekona θ mukona yekurudyi ndiyo chiyero chehurefu hwedivi rakapesana rekona θ nehurefu hwedivi riri pedyo rekona θ. Pamasvomhu, tan θ inoratidzwa seizvi:
\[ \tan \theta = \frac{\text{divi rakapesana rekona θ}}{\text{divi riri pedyo rekona θ}} \]
Kuti tinzwisise zviri nani pfungwa iyi, tichapfuura nemimwe mienzaniso yezvinetso uye tichakurukura mashandisirwo e tan θ.
Muenzaniso Mubvunzo 1: Kuverenga Tan θ
Tichipa bhora rerudyi rine kona θ panzvimbo A, uko divi repamusoro rekona θ rine urefu hwe3 cm uye divi riri pedyo rekona θ rine hurefu hwe4 cm. Verenga tan θ.
Mhinduro:
Kubva pamatambudziko ari pamusoro apa, tinoziva:
– Rutivi rwakapesana rwekona θ (rwakatarisana) = 3 cm
– Rutivi rwekona rwuri pedyo θ = 4 cm
Tichishandisa tsananguro ye tan θ, tinoverenga:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
\[ \tan \theta = \frac{3}{4} \]
Saka, tan θ = 0.75.
Zvichienderana nejometri, izvi zvinoreva kuti kune kona θ mutriangle, chiyero chehurefu hwedivi rakapesana nehurefu hwedivi riri pedyo i0.75.
Muenzaniso 2: Kushandisa Tan θ Kuverenga Kureba Kwemativi
Manera akasendamira pamadziro pakona yekukwirira ye30 degrees θ. Kureba kubva patsoka yemanera kusvika pamadziro mamita mashanu. Manera akasendamira pamadziro akareba sei?
Mhinduro:
Danho rekutanga, tinoyeuka tsananguro ye tan θ:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
Mumamiriro ezvinhu edambudziko iri:
– θ = madhigirii makumi matatu
– pedyo (daro kubva patsoka yemanera kusvika pamadziro) = mamita mashanu
– mhiri (kukwirira kwemanera kumadziro) = ???
Tinotanga taverenga\text{opposite)):
\[ \tan 30^\circ = \frac{\text{opposite}}{5} \]
Tinoziva kubva patafura yetrigonometric kuti:
\[ \tan 30^\circ = \frac{\sqrt{3}}{3} \]
Saka:
\[ \frac{\sqrt{3}}{3} = \frac{\text{opposite}}{5} \]
Wedzera mativi ese ari maviri ne5:
\[ \text{opposite} = 5 \cdot \frac{\sqrt{3}}{3} = \frac{5\sqrt{3}}{3} \]
Mhiri (kureba kwemanera kumadziro) ndekwekuti:
\[ \frac{5\sqrt{3}}{3} \inenge 2.89 \text{meters} \]
Saka, kureba kwemanera ndiko mamita mashanu.
Muenzaniso 3: Kuverenga Makona Uchishandisa Tan θ
Shongwe inokanda mumvuri wakareba mamita gumi nemaviri. Kana shongwe yacho yakareba mamita masere, kongiri yekukwirira kwezuva θ chii?
Mhinduro:
Mudambudziko iri, tinopiwa:
– Kureba kweshongwe (yakatarisana) = mamita masere
– Kureba kwemumvuri (pedyo) = mamita gumi nemaviri
Tinoshandisa tsananguro ye tan θ kuti tiwane θ:
\[ \tan \theta = \frac{8}{12} = \frac{2}{3} \]
Zvino tinowana θ ne equation:
\[ \theta = \tan^{-1} \left(\frac{2}{3}\right) \]
Tichitarisa patafura kana mukombuta kuti tione kukosha kwe inverse tangent, tinowana:
\[ \theta \approx 33.69^\circ \]
Saka, kona yekukwirira kwezuva inenge madhigirii 33.69.
Muenzaniso 4: Kushandisa Tan θ kuZvido zveNyika Yechokwadi
Chiratidziro chechiedza chakaiswa padanda remamita mana pamusoro pemotokari chakaiswa. Kana uchida kuisa siren inogona kuonekwa pakona yemadhigirii makumi mana nemashanu kubva pasi, verenga daro guru iro siren inogona kuonekwa.
Mhinduro:
Kubva pamubvunzo, zvinozivikanwa:
– Kureba kwedanda (rakatarisana) = mamita mana
– Angle θ = 45 degrees
Zvichienderana netsanangudzo ye tan θ:
\[ \tan 45^\circ = \frac{\text{opposite}}{\text{adjacent}} \]
Tinoziva kuti \(\tan 45^\circ = 1\), saka:
\[ 1 = \frac{4}{\text{adjacent}} \]
Saka:
\[ \text{adjacent} = 4 \text{meters} \]
Saka, daro riri kure zvikuru iro siren inogona kuonekwa naro i4 metres.
Mhedziso
Kubva mumienzaniso iri pamusoro, tinoona kuti tangent ye angle θ (\(\tan \theta\)) ipfungwa inobatsira zvikuru uye ine mashandisirwo akasiyana-siyana, kubva pakugadzirisa matambudziko ari nyore mumasvomhu kusvika pakushandiswa kwayo muzvinodiwa zvezuva nezuva, zvakaita sekuvaka nekufamba. Kunzwisisa kwakanaka kwepfungwa iyi kunogona kubatsira kugadzirisa matambudziko akasiyana-siyana anosanganisira kuenzanisa hurefu hwemativi ari mutriangle.
Kazhinji, tan θ, sechikamu chetrigonometry, haisi nyaya yakakosha chete mudzidzo yepamutemo asiwo chishandiso chinobatsira zvikuru muzvikamu zvakasiyana-siyana zvehupenyu chaihwo. Tinovimba kuti chinyorwa chino chinopa pfupiso yakajeka uye yakadzama yekuti ungashandisa sei tan θ kugadzirisa matambudziko ane chekuita nazvo.
