Cov Lus Nug Piv Txwv Sib Tham Txog Trigonometric Ratios Hauv Pyramids
Pendahuluuan
Qhov tseem ceeb, trigonometry yog ib ceg ntawm kev suav lej uas kawm txog kev sib raug zoo ntawm cov ces kaum thiab sab ntawm cov duab peb ceg. Trigonometric ratios yog qhov tseem ceeb hauv ntau qhov chaw xws li physics, engineering, thiab txawm tias architecture. Ib qho piv txwv zoo tshaj plaws ntawm kev siv trigonometry hauv architecture yog Pyramids ntawm Egypt. Hauv tsab xov xwm no, peb yuav tham txog trigonometric ratios siv cov piv txwv ntsig txog pyramids.
Kev Taw Qhia Txog Trigonometry hauv Pyramids
Cov pyramids hauv tebchaws Iziv, tshwj xeeb tshaj yog cov Pyramids ntawm Giza, yog cov qauv nto moo heev thiab tau yog lub ntsiab lus ntawm kev kawm los ntawm ntau tus kws lej thiab cov kws kes duab vajtse. Ib qho ntawm cov khoom tseem ceeb ntawm lub pyramid yog lub duab peb ceg. Cov duab peb ceg tuaj yeem pom hauv ob qho tib si profile saib thiab cross-sections.
Los ntawm lub pyramid, peb tuaj yeem nrhiav tau cov duab peb ceg uas muaj kaum sab xis, cov duab peb ceg sib npaug, thiab ntau lwm yam duab peb ceg. Kev siv trigonometry pab tau zoo heev rau kev txiav txim siab qhov ntev, qhov siab, thiab qhov nqes ntawm lub pyramid.
Piv txwv txog teeb meem
Lo lus nug 1: Xam qhov siab ntawm lub Pyramid
Xav tias lub pyramid muaj lub hauv paus ntev li 150 meters thiab ntug nqes (apothem) ntawm 130 meters. Lub pyramid siab npaum li cas?
Kev Sib Tham:
Hauv qhov teeb meem no, peb tau muab qhov hypotenuse thiab qhov ntev ntawm lub hauv paus ntawm lub pyramid. Txhawm rau xam qhov siab ntawm lub pyramid, peb tuaj yeem siv Pythagorean theorem. Lub pyramid tuaj yeem faib ua ob lub duab peb ceg.
1. Peb yuav tsum nrhiav ib nrab ntawm qhov ntev ntawm sab hauv qab kom ua tau ib daim duab peb sab xis.
\( \text{Ib nrab ntawm qhov ntev ntawm sab hauv paus} = \frac{150}{2} = 75 \text{ meters} \)
2. Peb paub tias:
\( a^2 + b^2 = c^2 \)
qhov twg \(a\) yog ib nrab ntawm qhov ntev ntawm sab hauv paus, \(b\) yog qhov siab ntawm lub pyramid, thiab \(c\) yog hypotenuse.
3. Muab cov lej ntxiv rau hauv kab zauv:
\( 75^2 + b^2 = 130^2 \)
4. Xam:
\( 5625 + b^2 = 16900 \)
\( b^2 = 16900 – 5625 \)
\( b^2 = 11275 \)
\( b = \sqrt{11275} \approx 106.2 \text{ meters} \)
Yog li, qhov siab ntawm lub pyramid yog li 106.2 meters.
Lo Lus Nug 2: Xam Lub Kaum Ntawm Qhov Ncua Ntawm Lub Pyramid
Lub kaum sab xis ntawm lub apothem ntawm lub pyramid mus rau lub hauv paus ntawm lub pyramid uas muaj sab hauv qab ntev li 150 meters thiab qhov siab li 106.2 meters yog dab tsi?
Kev Sib Tham:
Yuav nrhiav tau lub kaum sab xis (\(\theta\)) ntawm lub apothem mus rau lub hauv paus ntawm lub pyramid, peb siv tau lub function trigonometric, uas yog tangent (\(\tan\)).
1. Siv cov qauv \(\tan(\theta) = \frac{\text{height}}{\frac{\text{base}}{2}}).
2. Sau cov lej:
\( \tan(\theta) = \frac{106.2}{75} \)
3. Xam:
\( \tan(\theta) \approx 1.416 \)
4. Nrhiav lub kaum sab xis siv qhov inverse tangent (\(\tan^{-1}\)):
\( \theta = \tan^{-1}(1.416) \approx 54.14^\circ \)
Yog li, lub kaum sab xis ntawm lub apothem mus rau lub hauv paus ntawm lub pyramid yog li 54.14 degrees.
Lo lus nug 3: Xam qhov ntev ntawm Apothem nrog Sine thiab Cosine
Xav tias ib lub pyramid siab 120 meters thiab lub kaum sab xis ntawm lub pyramid mus rau lub hauv paus yog 55 degrees. Lub apothem ntev npaum li cas?
Kev Sib Tham:
Peb siv tau lub function sine los yog cosine los daws qhov teeb meem no.
1. Siv cosine los daws nws, nco ntsoov:
\( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
2. Rov kho qhov sib npaug rau Hypotenuse (apothem):
\( \text{Hypotenuse} = \frac{\text{Adjacent}}{\cos(\theta)} \)
3. Sau cov lej:
\( \text{Hypotenuse} = \frac{120}{\cos(55^\circ)} \)
4. Xam:
\( \cos(55^\circ) \approx 0.5736 \)
\( \text{Hypotenuse} = \frac{120}{0.5736} \approx 209.3 \text{ meters} \)
Yog li, qhov ntev ntawm lub apothem yog li 209.3 meters.
Xaus
Hauv cov teeb meem saum toj no, peb tau siv ntau yam piv txwv trigonometric los xam qhov siab, lub kaum sab xis, thiab qhov ntev ntawm lub pyramid. Nrog kev nkag siab txog trigonometry, peb tuaj yeem daws ntau yam teeb meem geometric uas yuav zoo li nyuaj thaum xub thawj siab. Trigonometry muab cov cuab yeej tseem ceeb rau kev nkag siab thiab daws cov teeb meem uas peb ntsib hauv lub ntiaj teb tiag tiag, tshwj xeeb tshaj yog hauv cov ntsiab lus architectural zoo li cov pyramids Iyiv.
