Imibuzo Yesibonelo Se-Trigonometry Nengxoxo
I-Trigonometry iyigatsha lezibalo elifunda ubudlelwano phakathi kwama-engeli nobude obuseceleni konxantathu. Ukuqonda imiqondo eyisisekelo ye-trigonometry kubalulekile ezinhlotsheni ezahlukene zokusebenza, kusukela ku-physics nobunjiniyela kuya ku-astronomy kanye ne-geography. Kulesi sihloko, sizochaza izibonelo eziningana zezinkinga kanye nezincazelo eziphelele ukusiza ukuqonda.
Isibonelo Umbuzo 1: Ukubala Izinhlangothi Zonxantathu Kusetshenziswa Indlela Yesine
Umbuzo:
Uma unikezwe unxantathu i-ABC, ene-engeli A = 30°, i-engeli B = 45°, kanye nohlangothi b = 10 cm. Bala ubude bohlangothi a.
Ingxoxo:
Sebenzisa umthetho wesono:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Faka amanani aziwayo:
\[ \frac{a}{\sin 30°} = \frac{10}{\sin 45°} \]
Siyazi ukuthi:
\[ \sin 30° = \frac{1}{2} \]
\[ \sin 45° = \frac{\sqrt{2}}{2} \]
Manje, faka la manani esikhundleni saleso sibalo:
\[ \frac{a}{\frac{1}{2}} = \frac{10}{\frac{\sqrt{2}}{2}} \]
Yenza kube lula lesi sibalo:
\[ 2a = \frac{10 \cdot 2}{\sqrt{2}} \]
\[ 2a = \frac{20}{\sqrt{2}} \]
Yenza kube lula ukuthi i-denominator ibe nkulu kangakanani:
\[ 2a = \frac{20 \cdot \sqrt{2}}{2} \]
\[ 2a = 10\sqrt{2} \]
\[ a = 5\sqrt{2} \]
Ngakho-ke, ubude bohlangothi u-a buyi-\( 5\sqrt{2} \) cm.
Isibonelo Umbuzo 2: Ukubala ama-Angles Usebenzisa Indlela ye-Cosine
Umbuzo:
Unxantathu unezinhlangothi a = 7 cm, b = 10 cm, kanye no-c = 5 cm. Thola usayizi we-engeli C.
Ingxoxo:
Sebenzisa umthetho we-cosines:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
Faka amanani aziwayo:
\[ 5^2 = 7^2 + 10^2 – 2 \cdot 7 \cdot 10 \cdot \cos C \]
Yenza kube lula lesi sibalo:
\[ 25 = 49 + 100 – 140 \cos C \]
\[25 = 149 – 140 \cos C \]
Hambisa u-149 uye ohlangothini lwesobunxele:
\[ 25 – 149 = -140 \cos C \]
\[ -124 = -140 \cos C \]
\[ \cos C = \frac{124}{140} \]
\[ \cos C = \frac{62}{70} \]
\[ \cos C = \frac{31}{35} \]
Sebenzisa i-calculator ukuthola i-\( \cos^{-1} \) (i-cosine ephambene):
\[ C \approx \cos^{-1}\left(\frac{31}{35}\right) \]
\[ C \cishe 25.84° \]
Ngakho-ke, ubukhulu be-engeli C cishe bungu-25.84°.
Isibonelo Umbuzo 3: Ukubala Ukuphakama kanye Nendawo Yonxantathu
Umbuzo:
Unxantathu unezinhlangothi ezimbili ubude a = 6 cm kanye no-b = 8 cm kanye ne-engeli phakathi kwazo, \(\theta\) = 60°. Bala ukuphakama nendawo yonxantathu.
Ingxoxo:
1. Ukubala Indawo Yonxantathu:
Sebenzisa ifomula yendawo kanxantathu:
\[ \text{Area} = \frac{1}{2} ab \sin \theta \]
Faka amanani aziwayo:
\[ \text{Area} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \sin 60° \]
Siyazi ukuthi:
\[ \sin 60° = \frac{\sqrt{3}}{2} \]
Ngakho-ke:
\[ \umbhalo{Indawo} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \umbhalo{Indawo} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \umbhalo{Indawo} = 24 \umbhalo \umbhalo \umbhalo{\umbhalo{3}}{2} \]
\[ \umbhalo{Indawo} = 12\sqrt{3} \]
Ngakho-ke, indawo yonxantathu ingu-\( 12\sqrt{3} \) cm².
2. Ukubala Ukuphakama Konxantathu Kusukela Esisekelo a:
Ukuze ubale ukuphakama konxantathu, chaza ukuphakama njengo-h bese usebenzisa ifomula yendawo:
\[ \text{Area} = \frac{1}{2} \times a \times h \]
\[ 12\sqrt{3} = \frac{1}{2} \times 6 \times h \]
\[ 12\sqrt{3} = amahora ama-3 \]
\[ h = \frac{12\sqrt{3}}{3} \]
\[ h = 4\sqrt{3} \]
Ngakho-ke, ukuphakama konxantathu kungu-\( 4\sqrt{3} \) cm.
Isibonelo Umbuzo 4: Ukunquma Izinhlangothi Zonxantathu Wesokudla
Umbuzo:
Kunxantathu ongakwesokudla, one-engeli engu-(\theta\) = 30° kanye nohlangothi oluhambisanayo lwe-engeli engu-5 cm, nquma ubude be-hypotenuse.
Ingxoxo:
Sebenzisa izilinganiso ze-trigonometric ze-engeli engu-30° kunxantathu ongakwesokudla:
\[ \sin \theta = \frac{\text{front}}{\text{hypotenuse}} \]
\[ \sin 30° = \frac{obverse}{hypotenuse} = \frac{5}{\text{hypotenuse}} \]
Siyazi ukuthi:
\[ \sin 30° = \frac{1}{2} \]
Ngakho-ke:
\[ \frac{1}{2} = \frac{5}{\text{hypotenuse}} \]
\[ \umbhalo{hypotenuse} = 10 \]
Ngakho-ke, ubude be-hypotenuse buyi-10 cm.
Isibonelo Umbuzo 5: Ukubala Ama-Angles Ngemisebenzi Ye-Trigonometric
Umbuzo:
Uma \( \tan \theta = \frac{3}{4} \), bala usayizi we-engeli \(\theta\).
Ingxoxo:
Ukusebenzisa ifomula ye-tangent ephambene:
\[ \theta = \tan^{-1} \left(\frac{3}{4}\right) \]
Ngosizo lwe-calculator:
\[ \theta \cishe 36.87° \]
Ngakho-ke, usayizi we-engeli \(\theta\) cishe u-36.87°.
Isiphetho
I-Trigonometry ingumqondo obanzi wezibalo osetshenziswa kabanzi emikhakheni eyahlukene. Ukuqonda indlela yokusebenzisa imithetho yama-sine, ama-cosine, neminye imisebenzi eyisisekelo kusenza sikwazi ukuxazulula izinkinga ezahlukahlukene ezihilela onxantathu nama-engeli. Ngezibonelo ezingenhla, kunethemba lokuthi abafundi bazothola ukuqonda okujulile futhi bakwazi ukukusebenzisa ezimweni ezahlukahlukene ezidinga ukuqonda i-trigonometry.
