Imibuzo Eyisibonelo Exoxa Ngezilinganiso Ze-Trigonometric
I-Trigonometry iyigatsha lezibalo elifunda ubudlelwano phakathi kwezinhlangothi nama-engeli kanxantathu. Isici esibalulekile se-trigonometry ukuqonda izilinganiso ze-trigonometric, okuhlanganisa i-sine (sin), i-cosine (cos), kanye ne-tangent (tan). Lesi sihloko sizoxoxa ngezibonelo eziningana zezilinganiso ze-trigonometric kanye nezincazelo zazo ezinemininingwane ukuze kube lula ukuqonda kahle imiqondo eyisisekelo ye-trigonometric.
Isibonelo Umbuzo 1: Ukubala Izindinganiso zeSin, Cos, kanye neTan
Umbuzo:
Umfundi ucelwa ukuthi athole amanani e-sin, cos, kanye ne-tan ye-engeli \( \theta \) kunxantathu ongakwesokudla, uma ubude bohlangothi oluphambene (engeli ephambene \(\theta \)) bungu-3 cm, ubude bohlangothi oluseduze (isisekelo) bungu-4 cm, kanti ubude be-hypotenuse (hypotenuse) bungu-5 cm.
Ingxoxo:
Isinyathelo sokuqala ukuhlonza uhlangothi ngalunye oluhambisana ne-engeli enikeziwe. Njengoba kwaziwa, kunxantathu ongakwesokudla onezinhlangothi ze-triple yakudala ye-Pythagorean (3, 4, 5), singasebenzisa ngqo amafomula ayisisekelo e-trigonometric.
– I-Sine (isono) isilinganiso esiphakathi kobude bohlangothi oluphambene kanye ne-hypotenuse.
\[
\sin(\theta) = \frac{\text{obverse side}}{\text{hypotenuse}} = \frac{3}{5}
\]
– I-Cosine (cos) isilinganiso esiphakathi kobude bohlangothi oluseduze kanye ne-hypotenuse.
\[
\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{4}{5}
\]
– I-Tangent (i-tan) isilinganiso esiphakathi kobude bohlangothi lwangaphambili nohlangothi.
\[
\tan(\theta) = \frac{\text{front side}}{\text{side side}} = \frac{3}{4}
\]
Ngakho-ke, amanani ezilinganiso ze-trigonometric ze-angle \(\theta\) yile:
\[
\sin(\theta) = \frac{3}{5}, \quad \cos(\theta) = \frac{4}{5}, \quad \tan(\theta) = \frac{3}{4}
\]
Isibonelo Umbuzo 2: Ukuthola Izinhlangothi Zonxantathu Kusetshenziswa Izilinganiso Ze-Trigonometric
Umbuzo:
Uma unikezwe unxantathu ongakwesokudla one-engeli \( \alpha = 30^\circ \). Uma ubude bohlangothi oluphambene lwe-engeli \( \alpha \) bungu-4 cm, nquma ubude bolunye uhlangothi.
Ingxoxo:
Ukusebenzisa amanani esilinganiso se-trigonometric e-angle \( 30^\circ \):
– \(\sin(30^\circ) = \frac{1}{2} \)
\[
\sin(30^\circ) = \frac{\text{front side}}{\text{hypotenuse}} = \frac{4}{\text{hypotenuse}}
\]
\[
\text{Hypotenuse} = \frac{4}{\sin(30^\circ)} = \frac{4}{\frac{1}{2}} = 8 \text{ cm}
\]
– \(\cos(30^\circ) = \frac{\sqrt{3}}{2} \)
\[
\cos(30^\circ) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\text{adjacent side}}{8}
\]
\[
\text{Side side} = 8 \cos(30^\circ) = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3} \text{ cm}
\]
Ngakho-ke, ubude bolunye uhlangothi buyi-\(4\sqrt{3}\) cm kanti i-hypotenuse ingu-8 cm.
Isibonelo 3: Ukusebenzisa i-Trigonometry kuma-Cartesian Coordinates
Umbuzo:
Nquma amanani e-sin, cos, kanye ne-tan ephuzwini \( P(3, 4) \) kuma-coordinates e-Cartesian uma iphuzu lakha i-engeli \(\theta\) ene-x-axis enhle ku-quadrant I.
Ingxoxo:
Okokuqala, bala ibanga lephuzu \(P(3, 4)\) kusukela ekuqaleni (O), okuyi-hypotenuse yonxantathu owakhiwe.
– I-hypotenuse (\(r\)) ingabalwa kusetshenziswa i-Pythagorean theorem:
\[
r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Ngemuva kwalokho, i-trigonometry yephuzu P ingabalwa kanje:
– I-Sine (isono) yisilinganiso esiphakathi kwe-ordinate (y) kanye ne-hypotenuse (r):
\[
\sin(\theta) = \frac{y}{r} = \frac{4}{5}
\]
– I-Cosine (cos) yisilinganiso esiphakathi kwe-abscissa (x) kanye ne-hypotenuse (r):
\[
\cos(\theta) = \frac{x}{r} = \frac{3}{5}
\]
– I-Tangent (tan) yisilinganiso esiphakathi kwe-ordinate (y) kanye ne-abscissa (x):
\[
\tan(\theta) = \frac{y}{x} = \frac{4}{3}
\]
Ngakho-ke, amanani e-trigonometric ephuzu P yile:
\[
\sin(\theta) = \frac{4}{5}, \quad \cos(\theta) = \frac{3}{5}, \quad \tan(\theta) = \frac{4}{3}
\]
Isibonelo Umbuzo 4: Ukusebenzisa Ubunikazi be-Trigonometric
Umbuzo:
Uma \( \sin(\theta) = \frac{3}{5} \), thola inani \( \cos(\theta) \) usebenzisa i-trigonometric identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
Ingxoxo:
Siqala ngobunikazi obuyisisekelo be-trigonometric:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
Uma kunikezwe \( \sin(\theta) = \frac{3}{5} \), bese kuthi:
\[
\sin^2(\theta) = \left(\frac{3}{5}\right)^2 = \frac{9}{25}
\]
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
\[
\frac{9}{25} + \cos^2(\theta) = 1
\]
\[
\cos^2(\theta) = 1 – \frac{9}{25} = \frac{25}{25} – \frac{9}{25} = \frac{16}{25}
\]
\[
\cos(\theta) = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}
\]
Inani le-\( \cos(\theta) \) lingaba lihle noma libe libi kuye ngokuthi i-quadrant i-engeli \(\theta\) ikhona. Kodwa kule nkinga ngaphandle kwencazelo ecacile ye-quadrant, sisebenzisa i-\(\cos(\theta) = \frac{4}{5}\) ye-quadrant I noma i-\( -\frac{4}{5}\) ye-quadrant II, III, noma i-IV.
Isiphetho
Ukuqonda izilinganiso ze-trigonometric kubalulekile ekuxazululeni izinkinga ezahlukahlukene ezihilela ama-engeli nobude kuma-triangles. Ngokuzijwayeza ukuxazulula izinkinga ezifana nalezi ezingenhla, ungathuthukisa ikhono lakho lokubala nokuqonda ubudlelwano phakathi kwezinhlangothi nama-engeli kuma-triangles. I-Trigonometry ayisetshenziswa kuphela ezibalweni ezihlanzekile kodwa futhi nasezindaweni ezahlukene zesayensi njenge-physics, ubunjiniyela, kanye ne-astronomy. Jabulela ukuzijwayeza!
