Su'aalo Tusaale ah oo Ka Hadlaya Saamiyada Trigonometric
Trigonometry waa laan xisaabeed oo barta xiriirka ka dhexeeya dhinacyada iyo xaglaha saddexagalka. Mid ka mid ah dhinacyada muhiimka ah ee trigonometry waa fahamka saamiga trigonometric, oo ay ku jiraan sine (sin), cosine (cos), iyo tangent (tan). Maqaalkani wuxuu ka hadli doonaa dhowr tusaale oo ku saabsan saamiga trigonometric iyo sharraxaaddooda faahfaahsan si loo fududeeyo faham dhammaystiran oo ku saabsan fikradaha aasaasiga ah ee trigonometric.
Su'aal Tusaale 1aad: Xisaabinta Qiimaha Dembiga, Koos, iyo Tan
Su'aal:
Ardayga waxaa la weydiisanayaa inuu helo qiimaha sin, cos, iyo tan ee xagasha \( \theta \) ee saddexagal midig, haddii dhererka dhinaca ka soo horjeeda (xagasha ka soo horjeeda \(\theta \)) uu yahay 3 cm, dhererka dhinaca ku xiga (saldhigga) uu yahay 4 cm, dhererka hypotenuse (hypotenuse) uu yahay 5 cm.
Dood:
Tallaabada ugu horreysa waa in la aqoonsado dhinac kasta oo khuseeya xagasha la bixiyay. Sida la og yahay, saddexagal midig oo leh dhinacyo saddexagal Pythagorean ah oo caadi ah (3, 4, 5), waxaan si toos ah u isticmaali karnaa qaacidooyinka trigonometric ee aasaasiga ah.
– Sine (sin) waa saamiga u dhexeeya dhererka dhinaca ka soo horjeeda iyo hypotenuse.
\[
\sin(\theta) = \frac{\text{obverse side}}{\text{hypotenuse}} = \frac{3}{5}
\]
– Cosine (cos) waa saamiga u dhexeeya dhererka dhinaca ku xiga iyo hypotenuse.
\[
\cos(\theta) = \frac{\text{dhinaca ku xiga}}{\text{hypotenuse}} = \frac{4}{5}
\]
– Tangent (tan) waa saamiga u dhexeeya dhererka dhinaca hore iyo dhinaca.
\[
\tan(\theta) = \frac{\text{dhinaca hore}}{\text{dhinaca dambe}} = \frac{3}{4}
\]
Sidaa darteed, qiimayaasha saamiga trigonometric ee xagasha \(\theta\) waa:
\[
\sin(\theta) = \frac{3}{5}, \quad \cos(\theta) = \frac{4}{5}, \quad \tan(\theta) = \frac{3}{4}
\]
Su'aal Tusaale 2: Helitaanka Dhinacyada Saddexagalka Adigoo Adeegsanaya Saamiyada Trigonometric
Su'aal:
Haddii la siiyo saddexagal midig oo leh xagal \( \alpha = 30^\circle \). Haddii dhererka dhinaca ka soo horjeeda ee xagasha \( \alpha \) uu yahay 4 cm, go'aami dhererka dhinaca kale.
Dood:
Isticmaalka qiimaha saamiga trigonometric ee xagasha \( 30^\circle \):
– \(\sin(30^\circ) = \frac{1}{2} \)
\[
\sin(30^\circ) = \frac{\text{front side}}{\text{hypotenuse}} = \frac{4}{\text{hypotenuse}}
\]
\[
\text{Hypotenuse} = \frac{4}{\sin(30^\circ)} = \frac{4}{\frac{1}{2}} = 8 \text{ cm}
\]
– \(\cos(30^\circ) = \frac{\sqrt{3}}{2} \)
\[
\cos(30^\circ) = \frac{\text{dhinaca ku xiga}}{\text{hypotenuse}} = \frac{\text{dhinaca ku xiga}}{8}
\]
\[
\qoraalka{Dhinaca} = 8 \cos(30^\circ) = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3} \qoraalka{ cm}
\]
Sidaas darteed, dhererka dhinaca kale waa \(4\sqrt{3}\) cm, halka hypotenuse-kuna yahay 8 cm.
Tusaale 3: Adeegsiga Trigonometry ee Isku-duwayaasha Cartesian
Su'aal:
Go'aami qiimaha dembiga, cos, iyo tan ee barta \( P(3, 4) \) ee isku-duwayaasha Cartesian haddii barta ay sameyso xagal \(\theta\) oo leh dhidibka x ee togan ee rubuc I.
Dood:
Marka hore, xisaabi masaafada barta \(P(3, 4)\) laga bilaabo asalka (O), kaas oo ah hypotenuse-ka saddexagalka la sameeyay.
– Hypotenuse (\(r\)) waxaa lagu xisaabin karaa iyadoo la adeegsanayo aragtida Pythagorean:
\[
r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Kadib, trigonometry-ga dhibicda P waxaa loo xisaabin karaa sidan soo socota:
– Sine (sin) waa saamiga u dhexeeya isku-dhafka (y) iyo hypotenuse (r):
\[
\sin(\theta) = \frac{y}{r} = \frac{4}{5}
\]
– Cosine (cos) waa saamiga u dhexeeya abscissa (x) iyo hypotenuse (r):
\[
\cos(\theta) = \frac{x}{r} = \frac{3}{5}
\]
– Tangent (tan) waa saamiga u dhexeeya isku-dhafka (y) iyo abscissa (x):
\[
\tan(\theta) = \frac{y}{x} = \frac{4}{3}
\]
Sidaa darteed, qiimayaasha trigonometric ee dhibicda P waa:
\[
\sin(\theta) = \frac{4}{5}, \quad \cos(\theta) = \frac{3}{5}, \quad \tan(\theta) = \frac{4}{3}
\]
Su'aal Tusaale ah 4: Adeegsiga Aqoonsiga Trigonometric
Su'aal:
Haddii \( \sin(\theta) = \frac{3}{5} \), hel qiimaha \( \cos(\theta) \) adoo isticmaalaya aqoonsiga trigonometric \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
Dood:
Waxaan ku bilaabaynaa aqoonsi aasaasi ah oo ku saabsan trigonometric:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
Marka la eego \( \sin(\theta) = \frac{3}{5} \), markaa:
\[
\sin^2(\theta) = \left(\frac{3}{5}\right)^2 = \frac{9}{25}
\]
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
\[
\frac{9}{25} + \cos^2(\theta) = 1
\]
\[
\cos^2(\theta) = 1 – \frac{9}{25} = \frac{25}{25} – \frac{9}{25} = \frac{16}{25}
\]
\[
\cos(\theta) = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}
\]
Qiimaha \( \cos(\theta) \) wuxuu noqon karaa mid togan ama taban iyadoo ku xiran rubuca xagasha \(\theta\) ku jirto. Laakiin dhibaatadan iyada oo aan lahayn qeexitaan cad oo rubuca ah, waxaan u isticmaalnaa \(\cos(\theta) = \frac{4}{5}\) rubuca I ama \( -\frac{4}{5}\) rubuca II, III, ama IV.
Gabagabo
Fahmidda saamiga trigonometric-ka ayaa muhiim u ah xallinta mashaakilaad kala duwan oo ku lug leh xaglaha iyo dhererka saddexagalka. Markaad ku tababarto xallinta mashaakilaadka sida kan kor ku xusan, waxaad horumarin kartaa awooddaada inaad xisaabiso oo aad fahamto xiriirka ka dhexeeya dhinacyada iyo xaglaha saddexagalka. Trigonometry-ga waxaa lagu dabaqaa ma aha oo kaliya xisaabta saafiga ah laakiin sidoo kale qaybaha kala duwan ee sayniska sida fiisigiska, injineernimada, iyo xiddigiska. Ku celcelin wanaagsan!
