Mienzaniso yeMibvunzo Inotsanangura Kushandiswa kweTrigonometric Ratios
Trigonometry ibazi remasvomhu rinoongorora hukama huripo pakati pemativi nemakona ematriangles. Kunzwisisa trigonometry kwakakosha nekuti kunowanzo shandiswa muzvikamu zvakasiyana-siyana, kusanganisira kuvaka, mainjiniya, nyeredzi, uye kunyangwe cryptography. Chinyorwa chino chichakurukura mimwe mienzaniso yezvinetso uye chichakurukura nezvazvo mukushandisa trigonometric ratios.
Pfungwa huru dzeTrigonometry
Tisati tapinda mumatambudziko emuenzaniso, ngationgororei dzimwe pfungwa huru mu trigonometry. Mu triangle yekurudyi, kune mabasa matatu makuru e trigonometry anowanzo shandiswa: sine, cosine, uye tangent.
– Sine (sin) yekona ihuremu hwehurefu hwerutivi rwakatarisana nekona nehurefu hwehypotenuse.
\[
\sin \theta = \frac{\text{obverse side}}{\text{hypotenuse}}
\]
– Cosine (cos) yekona ihuremu hwehurefu hwerutivi rwuri pedyo nekona nehurefu hwehypotenuse.
\[
\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}
\]
– Tangent (tan) yekona ihuremu hwehurefu hwerutivi rwakatarisana nekona nehurefu hwerutivi rwakatarisana nekona.
\[
\tan \theta = \frac{\text{front side}}{\text{side side}}
\]
Muenzaniso Mubvunzo 1: Kuverenga Kureba kweShongwe
Mubvunzo: Mucherechedzi anomira mamita makumi mashanu kubva pashongwe uye anoyera kona yekukwirira kweshongwe kusvika pamadhigirii makumi matatu. Sarudza kukwirira kweshongwe.
Mhinduro: Kuti tigadzirise dambudziko iri, tinogona kushandisa basa re tangent mu trigonometry. Sezvo tichiziva kona yekukwira uye daro rakatarwa kubva kumuoni kusvika pashongwe, tinogona kunyora:
\[
\tan 30^\circ = \frac{\text{tower height}}{\text{horizontal distance}}
\]
Tsiva tsika dzinozivikanwa:
\[
\tan 30^\circ = \frac{h}{50}
\]
Zvinozivikanwa kuti \(\tan 30^\circ = \frac{1}{\sqrt{3}}\), saka:
\[
\frac{1}{\sqrt{3}} = \frac{h}{50}
\]
Zvadaro kukwirira kweshongwe, \(h\), kunogona kuwanikwa nekuwedzera mativi ese e equation ne50:
\[
h = 50 \cdot \frac{1}{\sqrt{3}} = \frac{50}{\sqrt{3}} \inenge 28.87 \, \text{meter}
\]
Kureba kweshongwe iyi kunosvika mamita 28.87.
Muenzaniso Mubvunzo 2: Kuona Kureba Uchishandisa Cosine
Mubvunzo: Ngarava inoenda makiromita gumi kumabvazuva, yobva yachinja nzira nemadhigirii makumi matanhatu kuchamhembe yofamba makiromita gumi nemashanu. Sarudza daro kubva panotangira ngarava.
Kukurukurirana: Kuti tigadzirise dambudziko iri, tinogona kushandisa mutemo wecosine mu trigonometry. Kana tikaronga rwendo rwechikepe nenzira ye coordinate system, tinowana triangle ine mativi e 10 km ne 15 km, uye angle ye 60 degrees. Tinogona kushandisa mutemo wecosine kuti tiwane daro riri pakati penzvimbo yekutanga kwechikepe nenzvimbo yayo yekupedzisira.
\[
c^2 = a^2 + b^2 – 2ab \cos C
\]
Dimana:
– \( a = 10 \)
– \( b = 15 \)
– \( C = 60^\denderedzwa \)
Tsiva tsika dzinozivikanwa:
\[
c^2 = 10^2 + 15^2 – 2 \cdot 10 \cdot 15 \cdot \cos 60^\circ
\]
Tinoziva kuti \(\cos 60^\circ = 0.5\), saka:
\[
c^2 = 100 + 225 – 2 \cdot 10 \cdot 15 \cdot 0.5
\]
\[
c^2 = 100 + 225 – 150
\]
\[
c^2 = 175
\]
\[
c = \sqrt{175} \inenge 13.23 \, \text{km}
\]
Saka daro kubva panotangira ngarava rinosvika makiromita angangoita 13.23.
Muenzaniso Mubvunzo 3: Kushandisa Sine Kuona Mativi eTriangle
Mubvunzo: Muchikamu chetatu, mativi maviri ane 7 cm uye 10 cm, nekona pakati pawo ye 45 degrees. Verenga kureba kwerutivi rwechitatu rwechikamu chetatu.
Kukurukurirana: Tinogona kushandisa mutemo wezvivi kugadzirisa dambudziko iri. Mumutemo wezvivi, kune triangle ine mativi \(a\), \(b\), uye \(c\) uye kona \(C\) pakati pemativi \(a\) uye \(b\):
\[
\frac{c}{\sin C} = \frac{a}{\sin A} = \frac{b}{\sin B}
\]
Zvisinei, munyaya iyi, tinogona kushandisa zvakananga mutemo we cosines kuti zvinhu zvive nyore. Mutemo we cosines unoti:
\[
c^2 = a^2 + b^2 – 2ab \cos C
\]
Dimana:
– \( a = 7 \)
– \( b = 10 \)
– \( C = 45^\denderedzwa \)
\(\cos 45^\circ = \frac{\sqrt{2}}{2}\), zvekuti:
\[
c^2 = 7^2 + 10^2 – 2 \cdot 7 \cdot 10 \cdot \frac{\sqrt{2}}{2}
\]
\[
c^2 = 49 + 100 – 70\sqrt{2}
\]
\[
c^2 = 149 – 70\sqrt{2}
\]
Kuverenga kukosha kwe \( c \):
\[
c \approx \sqrt{149 – 70\sqrt{2}} \approx 5.97 \, \text{cm}
\]
Saka, kureba kwerutivi rwechitatu rwekatatu kanokwana masendimita 5.97.
Mhedziso
Trigonometry chishandiso chinobatsira zvikuru pakugadzirisa matambudziko akasiyana-siyana anosanganisira matriangles nema angles. Nekunzwisisa kwakanaka kwe sine, cosine, tangent, uye mitemo ye trigonometry, tinogona kugadzirisa matambudziko akawanda anoshanda. Chinyorwa chino chinokurukura mienzaniso yakati wandei yekushandisa trigonometric ratios, iyo yatinotarisira kuti ichabatsira vaverengi kunzwisisa zvakanyanya.
