Izibonelo Zemibuzo Nezingxoxo Ngezilinganiso Ezikhethekile Ze-Trigonometric
I-Trigonometry iyigatsha lezibalo elifunda ubudlelwano phakathi kwezinhlangothi nama-engeli kanxantathu. Umqondo owodwa obalulekile ku-trigonometry ukusetshenziswa kwama-engeli akhethekile ukuqonda izilinganiso ze-trigonometric. Ama-engeli akhethekile asetshenziswa kakhulu afaka phakathi u-0°, 30°, 45°, 60°, kanye no-90°. Lesi sihloko sizochaza izibonelo futhi sixoxe ngama-engeli akhethekile kuzilinganiso ze-trigonometric.
Isingeniso kuma-Angles Akhethekile
Ama-engeli akhethekile atholakala ekuhlaziyweni konxantathu abakhethekile, njenge-isosceles kanye nonxantathu abalinganayo. Nazi izindinganiso eziyisisekelo ze-trigonometric zama-engeli akhethekile okufanele uwakhumbule:
| I-engeli (θ) | Isono(θ) | I-Cos(θ) | I-Tan(θ) |
|———–|——–|——–|——–|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | – |
Ngokwazi la manani ayisisekelo, singaxazulula izinkinga ezahlukahlukene ezihilela izilinganiso ze-trigonometric zama-engeli akhethekile.
Imibuzo Eyisibonelo Nengxoxo
Ake sibheke eminye imibuzo eyisibonelo kanye nezingxoxo zayo:
Isibonelo Umbuzo 1
Umbuzo:
Bala inani le-\( \sin(30°) + \cos(60°) \).
Ingxoxo:
Sisebenzisa amanani ayisisekelo e-trigonometry ekhethekile ye-angle.
\[
\sin(30°) = \frac{1}{2}
\]
\[
\cos(60°) = \frac{1}{2}
\]
Ngakho-ke,
\[
\sin(30°) + \cos(60°) = \frac{1}{2} + \frac{1}{2} = 1
\]
Ngakho, \( \sin(30°) + \cos(60°) = 1 \).
Isibonelo Umbuzo 2
Umbuzo:
Nquma inani le-\( \tan(45°) \times \cos(45°) \).
Ingxoxo:
Sisebenzisa amanani avela etafuleni lama-angles akhethekile.
\[
\tan(45°) = 1
\]
\[
\cos(45°) = \frac{\sqrt{2}}{2}
\]
Ngakho-ke,
\[
\tan(45°) \times \cos(45°) = 1 \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}
\]
Ngakho-ke, \( \tan(45°) \times \cos(45°) = \frac{\sqrt{2}}{2} \).
Isibonelo Umbuzo 3
Umbuzo:
Uma \( \sin(θ) = \cos(θ) \), nquma inani \( θ \) ebangeni elisukela ku-0° kuya ku-90°.
Ingxoxo:
Kusukela ebudlelwaneni obuyisisekelo be-trigonometry:
\[
\sin(θ) = \cos(θ)
\]
Lokhu kusho ukuthi \( \tan(θ) = 1 \).
Inani le-\( θ \) eligcwalisa i-equation \( \tan(θ) = 1 \) lingu-45°.
Ngakho-ke, \( θ = 45° \).
Isibonelo Umbuzo 4
Umbuzo:
Bala inani le-\( \frac{\sin(30°)}{\cos(60°)} \).
Ingxoxo:
Sisebenzisa amanani avela etafuleni lama-angles akhethekile.
\[
\sin(30°) = \frac{1}{2}
\]
\[
\cos(60°) = \frac{1}{2}
\]
Ngakho-ke,
\[
\frac{\sin(30°)}{\cos(60°)} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1
\]
Ngakho, \( \frac{\sin(30°)}{\cos(60°)} = 1 \).
Isibonelo Umbuzo 5
Umbuzo:
Nquma inani le-\( \cos(30°) \times \tan(60°) \).
Ingxoxo:
Sisebenzisa amanani avela etafuleni lama-angles akhethekile.
\[
\cos(30°) = \frac{\sqrt{3}}{2}
\]
\[
\tan(60°) = \sqrt{3}
\]
Ngakho-ke,
\[
\cos(30°) \times \tan(60°) = \frac{\sqrt{3}}{2} \times \sqrt{3} = \frac{3}{2}
\]
Ngakho-ke, \( \cos(30°) \times \tan(60°) = \frac{3}{2} \).
Isibonelo Umbuzo 6
Umbuzo:
Thola inani le-\( 2 \sin(45°) \cos(45°) \).
Ingxoxo:
Sisebenzisa amanani avela etafuleni lama-angles akhethekile.
\[
\sin(45°) = \frac{\sqrt{2}}{2}
\]
\[
\cos(45°) = \frac{\sqrt{2}}{2}
\]
Ukuze,
\[
2 \sin(45°) \cos(45°) = 2 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} = 2 \times \frac{2}{4} = 1
\]
Ngakho, \( 2 \sono(45°) \cos(45°) = 1 \).
Isibonelo Umbuzo 7
Umbuzo:
Nquma inani le-\( \csc(30°) \).
Ingxoxo:
\( \csc(θ) \) iphambene nethi \( \sin(θ) \).
\[
\sin(30°) = \frac{1}{2}
\]
Ngakho-ke,
\[
\csc(30°) = \frac{1}{\sin(30°)} = \frac{1}{\frac{1}{2}} = 2
\]
Ngakho-ke, \( \csc(30°) = 2 \).
Isibonelo Umbuzo 8
Umbuzo:
Bala inani le-\( \cot(60°) \).
Ingxoxo:
\( \umbhede(θ) \) ukuphambene kokuthi \( \tan(θ) \).
\[
\tan(60°) = \sqrt{3}
\]
Ngakho-ke,
\[
\cot(60°) = \frac{1}{\tan(60°)} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\]
Ngakho-ke, \( \cot(60°) = \frac{\sqrt{3}}{3} \).
Isibonelo Umbuzo 9
Umbuzo:
Uma i-\( \theta \) iyi-engeli enenani le-trigonometric elingu-\( \sin(\theta) = \cos(45°) \), thola inani le-\( \theta \) ebangeni elisukela ku-0° kuya ku-90°.
Ingxoxo:
Kusukela etafuleni lama-engeli akhethekile:
\[
\cos(45°) = \frac{\sqrt{2}}{2}
\]
Ngakho-ke,
\[
\sin(\theta) = \frac{\sqrt{2}}{2}
\]
Kuyaziwa,
\[
\sin(45°) = \frac{\sqrt{2}}{2}
\]
Ngakho-ke, \( \theta = 45° \).
Isiphetho
Ukwazi ama-engeli akhethekile kanye namanani ayisisekelo e-trigonometric kubalulekile ekuqondeni imiqondo ye-trigonometry nasekuxazululeni izinkinga ezahlukahlukene zezibalo. Ngokuzijwayeza okufanele, ukubamba ngekhanda ithebula le-engeli elikhethekile kuba lula, futhi ukuxazulula izinkinga ze-trigonometry kuba ngokushesha futhi kuphumelele kakhudlwana.
Ekugcineni, lesi sihloko siveza izibonelo zezinkinga kanye nezingxoxo ezihilela ama-engeli akhethekile, okukusiza ukuthi uqonde ukuthi ungawasebenzisa kanjani amanani e-engeli akhethekile e-trigonometric ngendlela engokoqobo. Sithemba ukuthi lesi sihloko sibe usizo ekufundeni kwakho!
