Imibuzo Yesibonelo Exoxa Ngezilinganiso Ze-Trigonometric Kumaphiramidi
I-Pendahuluan
Empeleni, i-trigonometry iyigatsha lezibalo elifunda ubudlelwano phakathi kwama-engeli nezinhlangothi zonxantathu. Izilinganiso ze-trigonometric zibalulekile emikhakheni ehlukahlukene njengefiziksi, ubunjiniyela, ngisho nezakhiwo. Esinye sezibonelo ezinhle kakhulu zokusetshenziswa kwe-trigonometry ekwakhiweni kwezakhiwo yiPyramids of Egypt. Kulesi sihloko, sizoxoxa ngezilinganiso ze-trigonometric sisebenzisa izibonelo ezihlobene namaphiramidi.
Isingeniso ku-Trigonometry kumaPhiramidi
Amaphiramidi aseGibhithe, ikakhulukazi amaPhiramidi aseGiza, ayizakhiwo ezidumile kakhulu futhi abe yisihloko esifundwa yizazi zezibalo nabakhi abaningi. Esinye sezingxenye ezibalulekile zephiramidi unxantathu. Onxantathu bangatholakala kokubili emibonweni yephrofayili kanye nasezingxenyeni eziphambene.
Kuphiramidi, singathola onxantathu abagobile ngakwesokudla, onxantathu abalinganayo, kanye nezinye izimo ezihlukene ezingonxantathu. Ukusetshenziswa kwe-trigonometry kusiza kakhulu ekunqumeni ubukhulu, ukuphakama, kanye nokwehla kwephiramidi.
Isibonelo sezinkinga
Umbuzo 1: Ukubala Ukuphakama Kwephiramidi
"Ake sithi iphiramidi inobude besisekelo obungamamitha ayi-150 kanye nomkhawulo othambekile (i-apothem) wamamitha ayi-130. Ingakanani ukuphakama kwephiramidi?"
Ingxoxo:
Kule nkinga, sinikezwa i-hypotenuse kanye nobude besisekelo sephiramidi. Ukuze sibale ukuphakama kwephiramidi, singasebenzisa i-Pythagorean theorem. Iphiramidi ingahlukaniswa ibe onxantathu ababili abangakwesokudla.
1. Sidinga ukuthola ingxenye yobude bohlangothi lwesisekelo ukuze sakhe unxantathu wesokudla.
\( \text{Ingxenye yobude bohlangothi lwesisekelo} = \frac{150}{2} = 75 \text{meters} \)
2. Siyazi ukuthi:
\( a^2 + b^2 = c^2 \)
lapho \(a\) iyingxenye yobude bohlangothi lwesisekelo, \(b\) ukuphakama kwephiramidi, kanye \(c\) kuyi-hypotenuse.
3. Faka izinombolo ku-equation:
\( 75^2 + b^2 = 130^2 \)
4. Bala:
\( 5625 + b^2 = 16900 \)
\( b^2 = 16900 – 5625 \)
\( b^2 = 11275 \)
\( b = \sqrt{11275} \cishe kube ngu-106.2 \umbhalo{amamitha} \)
Ngakho-ke, ukuphakama kwephiramidi kungamamitha ayi-106.2.
Umbuzo 2: Ukubala i-Engela yokuthambekela kwephiramidi
"Iyini i-engeli yokuthambekela kwe-apothem yephiramidi eya phansi kwephiramidi enobude becala lesisekelo esingamamitha ayi-150 kanye nokuphakama okungamamitha ayi-106.2?"
Ingxoxo:
Ukuze sithole i-engeli yokuthambekela (\(\theta\)) ye-apothem esisekelweni sephiramidi, singasebenzisa umsebenzi we-trigonometric, okungukuthi i-tangent (\(\tan\)).
1. Sebenzisa ifomula \(\tan(\theta) = \frac{\text{height}}{\frac{\text{base}}{2}}\).
2. Faka izinombolo:
\( \tan(\theta) = \frac{106.2}{75} \)
3. Bala:
\( \tan(\theta) \cishe 1.416 \)
4. Thola i-engeli usebenzisa i-tangent ephambene (\(\tan^{-1}\)):
\( \theta = \tan^{-1}(1.416) \cishe 54.14^\circ \)
Ngakho-ke, i-engeli yokuthambekela kwe-apothem phansi kwephiramidi ingama-degrees angu-54.14.
Umbuzo 3: Ukubala Ubude be-Apothem nge-Sine ne-Cosine
"Ake sithi iphiramidi iphakeme ngamamitha angu-120 kanti i-engeli yokuthambekela kwephiramidi esisekelweni ingama-degrees angu-55. Ingakanani ubude bephiramidi?"
Ingxoxo:
Singasebenzisa umsebenzi we-sine noma we-cosine ukuxazulula le nkinga.
1. Sebenzisa i-cosine ukuyixazulula, ukhumbula:
\( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
2. Hlela kabusha i-equation ye-Hypotenuse (i-apothem):
\( \text{Hypotenuse} = \frac{\text{Adjacent}}{\cos(\theta)} \)
3. Faka izinombolo:
\( \text{Hypotenuse} = \frac{120}{\cos(55^\circ)} \)
4. Bala:
\( \cos(55^\circ) \cishe 0.5736 \)
\( \text{Hypotenuse} = \frac{120}{0.5736} \cishe kube ngu-209.3 \text{meters} \)
Ngakho-ke, ubude be-apothem bungaba amamitha angu-209.3.
Isiphetho
Ezinkingeni ezingenhla, sisebenzise izilinganiso ezahlukene ze-trigonometric ukuze sibale ukuphakama, i-engeli ye-slope, kanye nobude be-apothem bephiramidi. Ngokuqonda i-trigonometry, singaxazulula izinkinga ezahlukahlukene ze-geometric ezingase zibonakale ziyinkimbinkimbi ekuqaleni. I-Trigonometry inikeza ithuluzi elibalulekile lokuqonda nokuxazulula izinkinga esibhekana nazo ezweni langempela, ikakhulukazi ezimweni zokwakha njengephiramidi yaseGibhithe.
