Cov Lus Nug Piv Txwv Sib Tham Txog Peb Qhov Trigonometric Ratios
Trigonometry yog ib ceg ntawm kev suav lej uas kawm txog kev sib raug zoo ntawm qhov ntev thiab cov ces kaum hauv cov duab peb ceg. Ib qho ntawm cov ntsiab lus tseem ceeb hauv trigonometry yog cov piv txwv trigonometric: sine (sin), cosine (cos), thiab tangent (tan). Tsab xov xwm no yuav tham txog ntau qhov teeb meem piv txwv thiab kev sib tham txog cov piv txwv trigonometric kom yooj yim rau koj nkag siab.
1. Nkag Siab Txog Peb Qhov Sib Piv Trigonometric
Ua ntej tshaj plaws, cia peb nkag siab txog lub ntsiab lus ntawm sine, cosine thiab tangent.
- Tus sine (sin) ntawm lub kaum sab xis yog qhov sib piv ntawm qhov ntev ntawm sab nraud ntawm lub kaum sab xis rau qhov ntev ntawm hypotenuse ntawm lub duab peb sab.
- Tus cosine (cos) ntawm lub kaum sab xis yog qhov sib piv ntawm qhov ntev ntawm sab uas nyob ib sab ntawm lub kaum sab xis rau qhov ntev ntawm hypotenuse ntawm lub duab peb sab.
- Tus tangent (tan) ntawm lub kaum sab xis yog qhov sib piv ntawm qhov ntev ntawm sab nraud ntawm lub kaum sab xis rau qhov ntev ntawm sab uas nyob ib sab. Tus tangent kuj tseem tuaj yeem qhia tau tias yog tus lej ntawm sine thiab cosine: tan(θ) = sin(θ) / cos(θ).
2. Piv txwv cov lus nug thiab kev sib tham
Lo lus nug thib 1:
Muab ib daim duab peb sab uas muaj lub hypotenuse ntawm 10 cm thiab ib sab uas muaj lub kaum sab xis θ ntawm 6 cm. Txheeb xyuas cov nqi ntawm sin, cos, thiab tan ntawm lub kaum sab xis θ.
Kev Sib Tham:
Yuav kom nrhiav tau cov nqi ntawm sin(θ), cos(θ), thiab tan(θ), peb kuj yuav tsum paub qhov ntev ntawm sab uas nyob ib sab. Cia peb siv Pythagorean theorem los nrhiav qhov ntev ntawm sab uas nyob ib sab.
Pythagorean Theorem:
\[ a^2 + b^2 = c^2 \]
qhov twg c yog hypotenuse, a yog sab nraud ntawm lub kaum sab xis, thiab b yog sab uas nyob ib sab ntawm lub kaum sab xis.
Muab:
– Hypotenuse (c) = 10 cm
– Sab pem hauv ntej ntawm lub kaum sab xis θ (a) = 6 cm
Yog li ntawd:
\[ a^2 + b^2 = c^2 \]
\[ 6^2 + b^2 = 10^2 \]
\[ 36 + b^2 = 100 \]
\[ b^2 = 64 \]
\[ b = \sqrt{64} \]
\[ b = 8 \]
Yog li, qhov ntev ntawm sab (b) yog 8 cm.
Tom ntej no, peb tuaj yeem xam cov nqi ntawm sine, cosine, thiab tangent:
- Kev txhaum(θ) = Sab Nraud Sab / Hypotenuse
\[ \sin(θ) = \frac{6}{10} = 0.6 \]
- Cos(θ) = Sab Sab / Hypotenuse
\[ \cos(θ) = \frac{8}{10} = 0.8 \]
– Tan(θ) = Sab Pem Hauv Ntej / Sab Sab
\[ \tan(θ) = \frac{6}{8} = 0.75 \]
Lo lus nug thib 2:
Muab ib daim duab peb sab uas ntev ntawm sab nraud ntawm lub kaum sab xis α yog 5 cm thiab ntev ntawm sab nraud ntawm lub kaum sab xis α yog 12 cm. Nrhiav cov nqi ntawm sin, cos, thiab tan ntawm lub kaum sab xis α.
Kev Sib Tham:
Ib yam li hauv lo lus nug 1, cia peb siv Pythagorean theorem los nrhiav qhov ntev ntawm hypotenuse.
Muab:
– Sab pem hauv ntej ntawm lub kaum sab xis α (a) = 5 cm
– Sab ntawm lub kaum sab xis α (b) = 12 cm
Siv cov Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
\[ 5^2 + 12^2 = c^2 \]
\[ 25 + 144 = c^2 \]
\[ 169 = c^2 \]
\[ c = \sqrt{169} \]
\[ c = 13 \]
Yog li, qhov ntev ntawm hypotenuse (c) yog 13 cm.
Tom ntej no, peb tuaj yeem xam cov nqi ntawm sine, cosine, thiab tangent:
– Sin(α) = Sab/Hypotenuse
\[ \sin(α) = \frac{5}{13} \]
- Cos(α) = Sab Sab / Hypotenuse
\[ \cos(α) = \frac{12}{13} \]
- Tan(α) = Sab Pem Hauv Ntej / Sab
\[ \tan(α) = \frac{5}{12} \]
Lo lus nug thib 3:
Yog tias paub tias sin β = 0.6 thiab lub kaum sab xis β nyob hauv quadrant I, nrhiav cov nqi ntawm cos β thiab tan β.
Kev Sib Tham:
Muab qhov sin β = 0.6
Peb paub tias nyob rau hauv quadrant kuv tus nqi ntawm cos β kuj yog qhov zoo.
Siv cov cim trigonometric yooj yim:
\[ \sin^2(β) + \cos^2(β) = 1 \]
\[ (0.6)^2 + \cos^2(β) = 1 \]
\[ 0.36 + \cos^2(β) = 1 \]
\[ \cos^2(β) = 1 – 0.36 \]
\[ \cos^2(β) = 0.64 \]
\[ \cos(β) = \sqrt{0.64} \]
\[ \cos(β) = 0.8 \]
Tom ntej no, peb tuaj yeem xam tus nqi tangent:
\[ \tan(β) = \frac{\sin(β)}{\cos(β)} \]
\[ \tan(β) = \frac{0.6}{0.8} \]
\[ \tan(β) = 0.75 \]
3. Txoj kev
Lub tswv yim ntawm trigonometric triad (sin, cos, tan) yog qhov tseem ceeb thiab tseem ceeb rau kev nkag siab txog trigonometry feem ntau. Los ntawm kev nkag siab yuav ua li cas nrhiav thiab xam peb tus nqi no hauv ntau hom triangles, koj tuaj yeem daws ntau yam teeb meem trigonometry. Cov teeb meem uas tau tham saum toj no yuav tsum pab koj nkag siab yuav ua li cas siv cov tswv yim no hauv ntau qhov xwm txheej.
Kev nkag siab zoo txog trigonometry kuj tseem yuav ua rau nws yooj yim dua rau koj kawm cov ntsiab lus siab dua hauv lej thiab kev tshawb fawb, xws li calculus thiab physics. Tsis txhob yig los xyaum thiab nkag siab tob txog cov ntsiab lus no kom ncav cuag qib kev txawj ntse siab dua.
