Mienzaniso yeMibvunzo yeKukurukura nezveTrigonometric Ratios
Trigonometry ibazi remasvomhu rinoongorora hukama huripo pakati pemativi nemakona etriangle. Chimwe chinhu chakakosha chetrigonometry kunzwisisa zviyero zvetrigonometric, zvinosanganisira sine (sin), cosine (cos), uye tangent (tan). Chinyorwa chino chichakurukura mienzaniso yakati wandei yezvikamu zvetrigonometric uye tsananguro dzazvo dzakadzama kuti zvive nyore kunzwisisa pfungwa dzekutanga dzetrigonometric.
Muenzaniso Mubvunzo 1: Kuverenga Maitiro eChivi, Cos, uye Tan
Mubvunzo:
Mudzidzi anokumbirwa kuti awane kukosha kwe sin, cos, uye tan ye angle \( \theta \) mu right triangle, kana kureba kwedivi rakatarisana (angle yakatarisana \(\theta \)) kuri 3 cm, kureba kwedivi rakatarisana (base) kuri 4 cm, uye kureba kwe hypotenuse (hypotenuse) kuri 5 cm.
Kukurukurirana:
Danho rekutanga nderekuona divi rega rega rine chekuita nekona yakapihwa. Sezvinozivikanwa, mutriangle yekurudyi ine mativi ePythagorean triple yekare (3, 4, 5), tinogona kushandisa zvakananga maformula ekutanga etrigonometric.
– Sine (sin) inhamba iri pakati pehurefu hwedivi rakapesana ne hypotenuse.
\[
\sin(\theta) = \frac{\text{obverse side}}{\text{hypotenuse}} = \frac{3}{5}
\]
– Cosine (cos) inhamba iri pakati pehurefu hwedivi riri pedyo ne hypotenuse.
\[
\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{4}{5}
\]
– Tangent (tan) inhamba iri pakati pehurefu hwerutivi rwepamberi nerutivi.
\[
\tan(\theta) = \frac{\text{front side}}{\text{side side}} = \frac{3}{4}
\]
Saka, kukosha kwetrigonometric ratios ye angle \(\theta\) ndekwekuti:
\[
\sin(\theta) = \frac{3}{5}, \quad \cos(\theta) = \frac{4}{5}, \quad \tan(\theta) = \frac{3}{4}
\]
Muenzaniso Mubvunzo 2: Kutsvaga Mativi eTriangle Uchishandisa Trigonometric Ratios
Mubvunzo:
Kana kureba kwerutivi rwekurudyi rwekona kuri 4 cm, sarudza kureba kwerumwe rutivi.
Kukurukurirana:
Kushandisa trigonometric ratio values ye angle \( 30^\circ \):
– \(\chivi(30^\circ) = \frac{1}{2} \)
\[
\sin(30^\circ) = \frac{\text{front side}}{\text{hypotenuse}} = \frac{4}{\text{hypotenuse}}
\]
\[
\text{Hypotenuse} = \frac{4}{\sin(30^\circ)} = \frac{4}{\frac{1}{2}} = 8 \text{cm}
\]
– \(\cos(30^\circ) = \frac{\sqrt{3}}{2} \)
\[
\cos(30^\circ) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\text{adjacent side}}{8}
\]
\[
\text{Side side} = 8 \cos(30^\circ) = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3} \text{cm}
\]
Saka, kureba kwerumwe rutivi i \(4\sqrt{3}\) cm uye hypotenuse i8 cm.
Muenzaniso 3: Kushandisa Trigonometry muCartesian Coordinates
Mubvunzo:
Sarudza kukosha kwe sin, cos, uye tan kwepoindi \( P(3, 4) \) muCartesian coordinates kana poindi ikagadzira kona \(\theta\) ine x-axis yakanaka mu quadrant I.
Kukurukurirana:
Kutanga, verenga daro repoindi \(P(3, 4)\) kubva pakabva (O), inova hypotenuse yetriangle yakaumbwa.
– Hypotenuse (\(r\)) inogona kuverengerwa uchishandisa dzidziso yePythagorean:
\[
r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Zvadaro, trigonometry yepoindi P inogona kuverengerwa seinotevera:
– Sine (sin) inhamba iri pakati pe ordinate (y) ne hypotenuse (r):
\[
\chivi(\theta) = \frac{y}{r} = \frac{4}{5}
\]
– Cosine (cos) inhamba iri pakati peabscissa (x) nehypotenuse (r):
\[
\cos(\theta) = \frac{x}{r} = \frac{3}{5}
\]
– Tangent (tan) inhamba iri pakati pe ordinate (y) neabscissa (x):
\[
\tan(\theta) = \frac{y}{x} = \frac{4}{3}
\]
Saka, ma trigonometric values epoindi P ndeaya:
\[
\sin(\theta) = \frac{4}{5}, \quad \cos(\theta) = \frac{3}{5}, \quad \tan(\theta) = \frac{4}{3}
\]
Muenzaniso Mubvunzo 4: Kushandisa Trigonometric Identities
Mubvunzo:
Kana \( \sin(\theta) = \frac{3}{5} \), tsvaga kukosha kwe \( \cos(\theta) \) uchishandisa trigonometric identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
Kukurukurirana:
Tinotanga neruzivo rwekutanga rwe trigonometric:
\[
\chivi^2(\theta) + \cos^2(\theta) = 1
\]
Zvapiwa \( \sin(\theta) = \frac{3}{5} \), zvino:
\[
\sin^2(\theta) = \left(\frac{3}{5}\right)^2 = \frac{9}{25}
\]
\[
\chivi^2(\theta) + \cos^2(\theta) = 1
\]
\[
\frac{9}{25} + \cos^2(\theta) = 1
\]
\[
\cos^2(\theta) = 1 – \frac{9}{25} = \frac{25}{25} – \frac{9}{25} = \frac{16}{25}
\]
\[
\cos(\theta) = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}
\]
Kukosha kwe \( \cos(\theta) \) kunogona kuva kwakanaka kana kuti kusina kunaka zvichienderana ne quadrant iri mu angle \(\theta\) iri. Asi padambudziko iri pasina tsananguro yakajeka ye quadrant, tinoshandisa \(\cos(\theta) = \frac{4}{5}\) ye quadrant I kana \( -\frac{4}{5}\) ye quadrants II, III, kana IV.
Mhedziso
Kunzwisisa zviyero zvetrigonometric kwakakosha pakugadzirisa matambudziko akasiyana-siyana ane chekuita nemakona nehurefu mumatriangles. Nekudzidzira kugadzirisa matambudziko akaita seari pamusoro apa, unogona kuvandudza kugona kwako kuverenga nekunzwisisa hukama huripo pakati pemativi nematriangles mumatriangles. Trigonometry haingoshandiswi mumasvomhu chete asiwo muzvikamu zvakasiyana-siyana zvesainzi zvakaita sefizikisi, mainjiniya, uye nyeredzi. Unofara kudzidzira!
