Cov Lus Nug Piv Txwv Sib Tham Txog Kev Siv Trigonometric Ratios
Trigonometry yog ib ceg ntawm kev suav lej uas kawm txog kev sib raug zoo ntawm cov sab thiab cov ces kaum ntawm cov duab peb ceg. Kev nkag siab txog trigonometry yog qhov tseem ceeb vim tias nws feem ntau siv rau ntau qhov chaw, suav nrog kev tsim vaj tsev, kev tsim kho vaj tse, kev kawm txog hnub qub, thiab txawm tias cryptography. Tsab xov xwm no yuav tham txog ntau qhov teeb meem piv txwv thiab tham txog lawv hauv cov ntsiab lus ntawm kev siv cov piv txwv trigonometric.
Cov Ntsiab Lus Tseem Ceeb ntawm Trigonometry
Ua ntej peb nkag mus rau hauv cov teeb meem piv txwv, cia peb rov xyuas qee lub ntsiab lus tseem ceeb hauv trigonometry. Hauv daim duab peb sab xis, muaj peb lub luag haujlwm tseem ceeb trigonometric uas feem ntau siv: sine, cosine, thiab tangent.
- Tus sine (sin) ntawm lub kaum sab xis yog qhov sib piv ntawm qhov ntev ntawm sab uas nyob rau sab nraud ntawm lub kaum sab xis rau qhov ntev ntawm lub hypotenuse.
\[
\sin \theta = \frac{\text{obverse side}}{\text{hypotenuse}}
\]
- 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.
\[
\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}
\]
- Lub tangent (tan) 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 sab uas nyob ib sab ntawm lub kaum sab xis.
\[
\tan \theta = \frac{\text{front side}}{\text{side side}}
\]
Piv txwv lus nug 1: Xam qhov siab ntawm lub pej thuam
Lus Nug: Ib tug neeg soj ntsuam sawv ntawm 50 meters ntawm lub pej thuam thiab ntsuas lub kaum sab xis ntawm lub pej thuam kom yog 30 degrees. Txheeb xyuas qhov siab ntawm lub pej thuam.
Kev daws teeb meem: Txhawm rau daws qhov teeb meem no, peb tuaj yeem siv lub luag haujlwm tangent hauv trigonometry. Vim tias peb paub lub kaum sab xis thiab qhov deb ntawm kab rov tav ntawm tus neeg soj ntsuam mus rau lub pej thuam, peb tuaj yeem sau:
\[
\tan 30^\circ = \frac{\text{qhov siab ntawm lub pej thuam}}{\text{qhov deb ntawm kab rov tav}}
\]
Hloov cov nqi uas paub lawm:
\[
\tan 30^\circ = \frac{h}{50}
\]
Nws paub tias \(\tan 30^\circ = \frac{1}{\sqrt{3}}\), yog li ntawd:
\[
\frac{1}{\sqrt{3}} = \frac{h}{50}
\]
Ces qhov siab ntawm lub pej thuam, \(h\), tuaj yeem nrhiav tau los ntawm kev sib npaug ob sab ntawm qhov sib npaug los ntawm 50:
\[
h = 50 \cdot \frac{1}{\sqrt{3}} = \frac{50}{\sqrt{3}} \approx 28.87 \, \text{meter}
\]
Qhov siab ntawm lub pej thuam yog li ntawm 28.87 meters.
Piv txwv lus nug 2: Txheeb xyuas qhov deb siv Cosine
Lus Nug: Ib lub nkoj mus rau sab hnub tuaj 10 km, ces hloov txoj kev mus rau sab qaum teb 60 degrees thiab caij nkoj mus 15 km. Txheeb xyuas qhov deb ntawm qhov chaw pib mus rau lub nkoj.
Kev Sib Tham: Txhawm rau daws qhov teeb meem no, peb tuaj yeem siv txoj cai cosine hauv trigonometry. Yog tias peb kos lub nkoj txoj kev mus ncig hauv ib qho system coordinate, peb pom ib daim duab peb ceg nrog sab ntawm 10 km thiab 15 km, thiab lub kaum sab xis ntawm 60 degrees. Peb tuaj yeem siv txoj cai cosine los nrhiav qhov deb ntawm lub nkoj qhov chaw pib thiab nws qhov chaw kawg.
\[
c^2 = a^2 + b^2 – 2ab \cos C
\]
Qhov twg:
-\( ib = 10 \)
– \( b = 15 \)
– \( C = 60^\circ \)
Hloov cov nqi uas paub lawm:
\[
c^2 = 10^2 + 15^2 – 2 x 10 x 15 x cos 60^\circ
\]
Peb paub tias \(\cos 60^\circ = 0.5\), ces:
\[
c^2 = 100 + 225 – 2 x 10 x 15 x 0.5
\]
\[
c^2 = 100 + 225 – 150
\]
\[
c^2 = 175
\]
\[
c = \sqrt{175} \approx 13.23 \, \text{km}
\]
Yog li ntawd, qhov deb ntawm qhov chaw pib mus rau lub nkoj yog li 13.23 km.
Piv txwv lus nug 3: Siv Sine los txiav txim seb sab ntawm daim duab peb sab yog dab tsi
Lus Nug: Hauv ib daim duab peb ceg, ob sab yog 7 cm thiab 10 cm, nrog lub kaum sab xis ntawm lawv yog 45 degrees. Xam qhov ntev ntawm sab thib peb ntawm daim duab peb ceg.
Kev Sib Tham: Peb siv tau txoj cai ntawm cov sines los daws qhov teeb meem no. Hauv txoj cai ntawm cov sines, rau ib daim duab peb ceg uas muaj sab \(a\), \(b\), thiab \(c\) thiab lub kaum sab \(C\) ntawm sab \(a\) thiab \(b\):
\[
\frac{c}{\sin C} = \frac{a}{\sin A} = \frac{b}{\sin B}
\]
Txawm li cas los xij, hauv qhov no, peb tuaj yeem siv txoj cai ntawm cosines ncaj qha los ua kom yooj yim dua. Txoj cai ntawm cosines hais tias:
\[
c^2 = a^2 + b^2 – 2ab \cos C
\]
Qhov twg:
-\( ib = 7 \)
– \( b = 10 \)
– \( C = 45^\circ \)
\(\cos 45^\circ = \frac{\sqrt{2}}{2}\), yog li ntawd:
\[
c^2 = 7^2 + 10^2 – 2\cdot 7\cdot 10\cdot \frac{\sqrt{2}}{2}
\]
\[
c^2 = 49 + 100 – 70\sqrt{2}
\]
\[
c^2 = 149 – 70\sqrt{2}
\]
Xam tus nqi ntawm \( c \):
\[
c \kwv yees li 149 – 70\kwv yees li 5.97 cm
\]
Yog li, qhov ntev ntawm sab thib peb ntawm daim duab peb sab yog li 5.97 cm.
Xaus
Trigonometry yog ib qho cuab yeej siv tau zoo heev rau kev daws ntau yam teeb meem uas cuam tshuam txog cov duab peb ceg thiab cov ces kaum. Yog tias peb nkag siab zoo txog sine, cosine, tangent, thiab cov cai ntawm trigonometry, peb tuaj yeem daws tau ntau yam teeb meem. Tsab xov xwm no tham txog ntau qhov piv txwv ntawm kev siv cov piv txwv trigonometric, uas peb vam tias yuav pab cov neeg nyeem ntawv nkag siab ntxiv.
