Cov Lus Nug Piv Txwv Sib Tham Txog Trigonometric Ratios
Trigonometry yog ib ceg ntawm kev lej uas cuam tshuam nrog kev sib raug zoo ntawm qhov ntev thiab cov ces kaum hauv cov duab peb ceg. Trigonometric ratios ua lub luag haujlwm tseem ceeb hauv ntau qhov chaw xws li engineering, physics, astronomy, thiab txawm tias lub neej txhua hnub. Tsab xov xwm no yuav tshuaj xyuas ntau qhov piv txwv teeb meem thiab lawv cov kev daws teeb meem los pab koj nkag siab txog cov ntsiab lus tseem ceeb ntawm trigonometric ratios.
Cov Piv Txwv Trigonometric hauv Cov Duab Peb Sab Xis
Cia peb pib los ntawm kev nkag siab txog cov piv txwv yooj yim hauv daim duab peb sab xis. Cov piv txwv no hu ua sine (sin), cosine (cos), thiab tangent (tan). Hauv daim duab peb sab xis, muaj peb sab tseem ceeb uas peb yuav tsum txheeb xyuas:
1. Sab Nraud: Sab uas nyob rau sab nraud ntawm lub kaum sab xis uas peb tab tom xav txog.
2. Sab Sib Ze: Sab uas nyob ib sab ntawm lub kaum sab xis uas peb tab tom xav txog.
3. Hypotenuse: Sab ntev tshaj plaws hauv daim duab peb sab uas tig rau sab xis.
Piv txwv lus nug 1
Muab ib daim duab peb ceg uas muaj lub kaum sab xis θ, qhov twg sab nraud rau lub kaum sab xis θ muaj qhov ntev ntawm 3 units, sab uas nyob ib sab muaj qhov ntev ntawm 4 units, thiab lub hypotenuse muaj qhov ntev ntawm 5 units. Nrhiav cov nqi ntawm sin, cos, thiab tan ntawm lub kaum sab xis θ.
Kev daws teeb meem:
Txhawm rau nrhiav cov nqi sine, cosine, thiab tangent ntawm lub kaum sab xis θ, peb siv cov qauv piv txwv trigonometric yooj yim:
– Sine (kev txhaum) θ = Pem Hauv Ntej Sab / Hypotenuse
\[ \sin θ = \frac{3}{5} \]
– Cosine (cos) θ = Sab Sab / Hypotenuse
\[ \cos θ = \frac{4}{5} \]
– Tangent (tan) θ = Sab Pem Hauv Ntej / Sab
\[ \tan θ = \frac{3}{4} \]
Kev Sib Raug Zoo ntawm Trigonometric ntawm Lwm Lub kaum sab xis
Cov piv ntawm trigonometric kuj siv tau rau lwm cov ces kaum hauv ntau hom duab peb ceg, suav nrog cov duab peb ceg sib npaug thiab cov duab peb ceg ib txwm. Nws yog ib qho tseem ceeb kom nkag siab txog cov cai yooj yim xws li Txoj Cai Sine thiab Txoj Cai Cosine, uas siv rau cov duab peb ceg uas tsis yog cov duab peb ceg sab xis.
Piv txwv lus nug 2
Muab ib daim duab peb ceg ABC nrog sab a = 7 cm, b = 24 cm, thiab lub kaum sab C = 90°. Nrhiav qhov ntev ntawm sab c thiab cov nqi ntawm cov kaum sab A thiab B.
Kev daws teeb meem:
Vim tias lub kaum sab xis C yog lub kaum sab xis, peb siv tau Pythagorean Theorem los nrhiav qhov ntev ntawm sab c (lub hypotenuse):
\[ c = \sqrt{a^2 + b^2} \]
\[ c = \sqrt{7^2 + 24^2} \]
\[ c = \sqrt{49 + 576} \]
\[ c = \sqrt{625} \]
\[ c = 25 \, \text{cm} \]
Tom ntej no, txhawm rau txiav txim siab tus nqi ntawm cov ces kaum A thiab B, peb tuaj yeem siv cov piv txwv trigonometric yooj yim.
Rau lub kaum sab xis A, peb siv cosine:
\[ \cos A = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{24}{25} \]
\[ A = \cos^{-1} \left(\frac{24}{25}\right) \]
Rau lub kaum sab xis B, peb siv sine:
\[ \sin B = \frac{\text{obverse side}}{\text{hypotenuse}} = \frac{24}{25} \]
\[ B = \sin^{-1} \left(\frac{24}{25}\right) \]
Vim tias nyob rau hauv ib daim duab peb sab xis qhov sib sau ua ke ntawm cov ces kaum uas tsis yog lub ces kaum sab xis yuav tsum yog 90°:
\[ A + B = 90° \]
\[ A = 90° – B \]
Txoj Cai Sine
Txoj cai sine tau hais raws li nram no:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Piv txwv lus nug 3
Muab ib daim duab peb ceg uas muaj sab a = 8 cm, sab b = 15 cm, thiab lub kaum sab C = 60°. Nrhiav lub kaum sab A thiab qhov ntev ntawm sab c siv Txoj Cai ntawm Sines.
Kev daws teeb meem:
Ua ntej, siv Txoj Cai ntawm Sines los nrhiav ib lub ces kaum ntawm lwm lub ces kaum (piv txwv li, lub kaum sab xis A):
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
\[ \frac{8}{\sin A} = \frac{c}{\sin 60°} \]
Peb yuav tsum xam tus nqi ntawm c ua ntej. Vim tias lub kaum sab xis C yog 60°, peb siv tau txoj cai Cosine:
\[ c^2 = a^2 + b^2 – 2ab \cos(C) \]
\[ c^2 = 8^2 + 15^2 – 2 \cdot 8 \cdot 15 \cdot \cos(60°) \]
\[ c^2 = 64 + 225 – 240 ∑0.5 \]
\[ c^2 = 289 – 120 \]
\[ c^2 = 169 \]
\[ c = 13 \, \text{cm} \]
Tam sim no, peb tuaj yeem nrhiav lub kaum sab xis A:
\[ \frac{8}{\sin A} = \frac{13}{\sin 60°} \]
Vim tias \(\sin 60° = \sqrt{3}/2\):
\[ \frac{8}{\sin A} = \frac{13}{\sqrt{3}/2} \]
\[ \frac{8}{\sin A} = \frac{26}{\sqrt{3}} \]
\[ 8 \cdot \sqrt{3} = 26 \sin A \]
\[ \sin A = \frac{8 \sqrt{3}}{26} \]
\[ \sin A = \frac{4 \sqrt{3}}{13} \]
Thaum kawg, siv cov inverse sine:
\[ A = \sin^{-1}\left(\frac{4 \sqrt{3}}{13}\right) \]
Txoj Cai Cosine
Txoj cai cosine hais tias:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
Piv txwv lus nug 4
Muab ib daim duab peb ceg uas muaj sab a = 9 cm, sab b = 12 cm, thiab sab c = 15 cm. Nrhiav lub kaum sab C.
Kev daws teeb meem:
Yuav nrhiav tau lub kaum sab xis C, peb siv txoj cai Cosine:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
\[ 15^2 = 9^2 + 12^2 – 2 \cdot 9 \cdot 12 \cdot \cos C \]
\[ 225 = 81 + 144 – 216 \cos C \]
\[ 225 = 225 – 216 \cos C \]
\[ 0 = -216 \cos C \]
\[ \cosC = 0 \]
Los ntawm no peb paub tias lub kaum sab xis C yog 90 ° vim cos 90 ° = 0.
Nov yog qee cov piv txwv teeb meem los pab koj nkag siab txog trigonometric ratios. Kev xyaum ua ntxiv nrog ntau yam teeb meem yuav pab txhawb koj txoj kev nkag siab txog cov ntsiab lus no. Peb vam tias tsab xov xwm no tau pab tau koj txoj kev kawm!
