Muenzaniso weTrigonometry Mibvunzo neKukurukurirana
Trigonometry ibazi remasvomhu rinoongorora hukama huripo pakati pemakona nehurefu hwemativi mumatriangles. Kunzwisisa pfungwa huru dzetrigonometry kwakakosha mukushandiswa kwakasiyana-siyana, kubva pafizikisi neinjiniya kusvika kuastronomy nejogirafi. Muchinyorwa chino, tichatsanangura mienzaniso yakati wandei yezvinetso uye tsananguro dzakakwana dzekubatsira kunzwisisa.
Muenzaniso Mubvunzo 1: Kuverenga Mativi eTriangle Uchishandisa Nzira yeSine
Mubvunzo:
Kana wapiwa triangle ABC, ine kona A = 30°, kona B = 45°, uye divi b = 10 cm. Verenga kureba kwedivi a.
Kukurukurirana:
Shandisa mutemo wezvivi:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Isa kukosha kunozivikanwa:
\[ \frac{a}{\sin 30°} = \frac{10}{\sin 45°} \]
Tinoziva kuti:
\[ \sin 30° = \frac{1}{2} \]
\[ \sin 45° = \frac{\sqrt{2}}{2} \]
Zvino, chinja maitiro aya mu equation:
\[ \frac{a}{\frac{1}{2}} = \frac{10}{\frac{\sqrt{2}}{2}} \]
Nyoresa equation:
\[ 2a = \frac{10 \cdot 2}{\sqrt{2}} \]
\[ 2a = \frac{20}{\sqrt{2}} \]
Rondedzera chikonzero che denominator:
\[ 2a = \frac{20 \cdot \sqrt{2}}{2} \]
\[ 2a = 10\sqrt{2} \]
\[ a = 5\sqrt{2} \]
Saka, kureba kwerutivi a i \( 5\sqrt{2} \) cm.
Muenzaniso Mubvunzo 2: Kuverenga Makona Uchishandisa Nzira yeCosine
Mubvunzo:
Katatu kane mativi a = 7 cm, b = 10 cm, uye c = 5 cm. Sarudza saizi yekona C.
Kukurukurirana:
Shandisa mutemo we cosines:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
Isa kukosha kunozivikanwa:
\[ 5^2 = 7^2 + 10^2 – 2 \cdot 7 \cdot 10 \cdot \cos C \]
Nyoresa equation:
\[ 25 = 49 + 100 – 140 \cos C \]
\[25 = 149 – 140 \cos C \]
Enda 149 kuruboshwe:
\[ 25 – 149 = -140 \cos C \]
\[ -124 = -140 \cos C \]
\[ \cos C = \frac{124}{140} \]
\[ \cos C = \frac{62}{70} \]
\[ \cos C = \frac{31}{35} \]
Shandisa karukureta kuti uwane \( \cos^{-1} \) (inverse cosine):
\[ C \approx \cos^{-1}\left(\frac{31}{35}\right) \]
\[ C \inenge 25.84° \]
Saka, saizi yekona C inenge 25.84°.
Muenzaniso Mubvunzo 3: Kuverenga Kureba uye Nzvimbo yeTriangle
Mubvunzo:
Katatu ine mativi maviri akareba a = 6 cm uye b = 8 cm ine kona pakati pawo, \(\theta\) = 60°. Verenga kukwirira nenzvimbo yekatatu.
Kukurukurirana:
1. Kuverenga Nzvimbo yeTriangle:
Shandisa fomura yenzvimbo yetriangle:
\[ \text{Area} = \frac{1}{2} ab \sin \theta \]
Isa tsika dzinozivikanwa:
\[ \text{Area} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \sin 60° \]
Tinoziva kuti:
\[ \sin 60° = \frac{\sqrt{3}}{2} \]
Saka:
\[ \text{Area} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{Area} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{Area} = 24 \cdot \frac{\sqrt{3}}{2} \]
\[ \chinyorwa{Nzvimbo} = 12\sqrt{3} \]
Saka, nzvimbo yetriangle i \( 12\sqrt{3} \) cm².
2. Kuverenga Kureba kweTriangle kubva paBase a:
Kuti uverenge kukwirira kwetriangle, ratidza kukwirira sa h uye shandisa fomura yenzvimbo:
\[ \text{Area} = \frac{1}{2} \times a \times h \]
\[ 12\sqrt{3} = \frac{1}{2} \kawa 6 \kawa hr \]
\[ 12\sqrt{3} = maawa matatu \]
\[ h = \frac{12\sqrt{3}}{3} \]
\[ h = 4\sqrt{3} \]
Saka, kukwirira kwetriangle i \( 4\sqrt{3} \) cm.
Muenzaniso Mubvunzo 4: Kuziva Mativi eTriangle Yekurudyi
Mubvunzo:
Muchikamu chetriangle yekurudyi, nekona \(\theta\) = 30° uye divi rakafanana rekona \(\theta\) riri 5 cm, sarudza kureba kwehypotenuse.
Kukurukurirana:
Shandisa zviyero zve trigonometric kuti uwane 30° angle mu right triangle:
\[ \sin \theta = \frac{\text{front}}{\text{hypotenuse}} \]
\[ \sin 30° = \frac{obverse}{hypotenuse} = \frac{5}{\text{hypotenuse}} \]
Tinoziva kuti:
\[ \sin 30° = \frac{1}{2} \]
Saka:
\[ \frac{1}{2} = \frac{5}{\text{hypotenuse}} \]
\[ \text{hypotenuse} = 10 \]
Saka, kureba kwe hypotenuse ndeye 10 cm.
Muenzaniso Mubvunzo 5: Kuverenga Makona neMabasa eTrigonometric
Mubvunzo:
Kana \( \tan \theta = \frac{3}{4} \), verenga saizi yekona \(\theta\).
Kukurukurirana:
Kushandisa fomura ye inverse tangent:
\[ \theta = \tan^{-1} \left(\frac{3}{4}\right) \]
Nerubatsiro rwekarukureta:
\[ \theta \inenge 36.87° \]
Saka, saizi yekona \(\theta\) inenge 36.87°.
Mhedziso
Trigonometry ipfungwa yakakura yemasvomhu ine mashandisirwo akasiyana-siyana. Kunzwisisa mashandisirwo emitemo yesines, cosines, uye mamwe mabasa ekutanga kunotibvumira kugadzirisa matambudziko akasiyana-siyana ane chekuita nematriangles nemakona. Kuburikidza nemienzaniso iri pamusoro, zvinotarisirwa kuti vaverengi vachanzwisisa zvakadzama uye vagokwanisa kuishandisa mumamiriro akasiyana-siyana anoda kunzwisisa trigonometry.
