Imibuzo Eyisibonelo Exoxa Ngokusetshenziswa Kwezilinganiso Ze-Trigonometric
I-Trigonometry iyigatsha lezibalo elifunda ubudlelwano phakathi kwezinhlangothi nama-engeli kanxantathu. Ukuqonda i-trigonometry kubalulekile ngoba ivame ukusetshenziswa emikhakheni ehlukahlukene, okuhlanganisa ukwakheka kwezakhiwo, ubunjiniyela, izinkanyezi, ngisho ne-cryptography. Lesi sihloko sizoxoxa ngezinkinga eziningana eziyizibonelo futhi sizixoxe ngomongo wokusebenzisa izilinganiso ze-trigonometric.
Imiqondo Eyisisekelo Ye-Trigonometry
Ngaphambi kokuthi singene ezinkingeni zesibonelo, ake sibukeze imiqondo eyisisekelo ku-trigonometry. Kunxantathu ongakwesokudla, kunemisebenzi emithathu eyinhloko ye-trigonometric evame ukusetshenziswa: i-sine, i-cosine, kanye ne-tangent.
– I-sine (isono) ye-engeli isilinganiso sobude bohlangothi oluphambene ne-engeli nobude be-hypotenuse.
\[
\sin \theta = \frac{\text{obverse side}}{\text{hypotenuse}}
\]
– I-cosine (cos) ye-engeli iyisilinganiso sobude bohlangothi oluseduze ne-engeli nobude be-hypotenuse.
\[
\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}
\]
– I-tangent (tan) ye-engeli isilinganiso sobude bohlangothi oluphambene ne-engeli nobude bohlangothi oluseduze ne-engeli.
\[
\tan \theta = \frac{\text{front side}}{\text{side side}}
\]
Isibonelo Umbuzo 1: Ukubala Ukuphakama Kombhoshongo
Umbuzo: Umbukeli umi amamitha angu-50 ukusuka embhoshongweni futhi ulinganisa i-engeli yokuphakama kwengxenye ephezulu yombhoshongo ibe ngama-degree angu-30. Thola ukuphakama kombhoshongo.
Isixazululo: Ukuze sixazulule le nkinga, singasebenzisa umsebenzi we-tangent ku-trigonometry. Njengoba sazi i-engeli yokuphakama kanye nebanga eliqondile ukusuka kumbukeli kuya embhoshongweni, singabhala:
\[
\tan 30^\circ = \frac{\text{tower height}}{\text{horizontal distance}}
\]
Faka amanani aziwayo esikhundleni sawo:
\[
\tan 30^\circ = \frac{h}{50}
\]
Kuyaziwa ukuthi \(\tan 30^\circ = \frac{1}{\sqrt{3}}\), ukuze:
\[
\frac{1}{\sqrt{3}} = \frac{h}{50}
\]
Bese ukuphakama kombhoshongo, \(h\), kungatholakala ngokuphindaphinda izinhlangothi zombili zesibalo ngo-50:
\[
h = 50 \cdot \frac{1}{\sqrt{3}} = \frac{50}{\sqrt{3}} \cishe kube ngu-28.87 \, \text{meter}
\]
Ukuphakama kombhoshongo kungamamitha angu-28.87.
Isibonelo Umbuzo 2: Ukunquma Ibanga Usebenzisa i-Cosine
Umbuzo: Umkhumbi uya empumalanga ngamakhilomitha ayi-10, bese ushintsha indlela ngama-degree angu-60 enyakatho bese uhamba ngamakhilomitha ayi-15. Thola ibanga ukusuka endaweni yokuqala ukuya emkhunjini.
Ingxoxo: Ukuze sixazulule le nkinga, singasebenzisa umthetho we-cosine ku-trigonometry. Uma sihlela uhambo lomkhumbi ohlelweni lokuhlanganisa, sithola unxantathu onezinhlangothi ezingama-10 km kanye nama-15 km, kanye ne-engeli engama-degrees angu-60. Singasebenzisa umthetho we-cosine ukuthola ibanga phakathi kwendawo yokuqala komkhumbi nendawo yawo yokugcina.
\[
c^2 = a^2 + b^2 – 2ab \cos C
\]
Kuphi:
– \( a = 10 \)
– \( b = 15 \)
– \( C = 60^\circ \)
Faka amanani aziwayo esikhundleni sawo:
\[
c^2 = 10^2 + 15^2 – 2 \cdot 10 \cdot 15 \cdot \cos 60^\circ
\]
Siyazi ukuthi \(\cos 60^\circ = 0.5\), bese kuba:
\[
c^2 = 100 + 225 – 2 \cdot 10 \cdot 15 \cdot 0.5
\]
\[
c^2 = 100 + 225 – 150
\]
\[
c^2 = 175
\]
\[
c = \sqrt{175} \cishe 13.23 \, \umbhalo{km}
\]
Ngakho-ke ibanga ukusuka endaweni yokuqala ukuya emkhunjini lingamakhilomitha ayi-13.23.
Isibonelo Umbuzo 3: Ukusebenzisa i-Sine Ukunquma Izinhlangothi Zonxantathu
Umbuzo: Kunxantathu, izinhlangothi ezimbili zingama-7 cm kanye nama-10 cm, kanti i-engeli phakathi kwazo ingu-45 degrees. Bala ubude bohlangothi lwesithathu lonxantathu.
Ingxoxo: Singasebenzisa umthetho we-sines ukuxazulula le nkinga. Emthethweni we-sines, unxantathu onezinhlangothi \(a\), \(b\), kanye \(c\) kanye ne-engeli \(C\) phakathi kwezinhlangothi \(a\) kanye \(b\):
\[
\frac{c}{\sin C} = \frac{a}{\sin A} = \frac{b}{\sin B}
\]
Kodwa-ke, kulokhu, singasebenzisa ngqo umthetho wama-cosine ukuze senze izinto zibe lula. Umthetho wama-cosine uthi:
\[
c^2 = a^2 + b^2 – 2ab \cos C
\]
Kuphi:
– \( a = 7 \)
– \( b = 10 \)
– \( C = 45^\circ \)
\(\cos 45^\circ = \frac{\sqrt{2}}{2}\), ukuze:
\[
c^2 = 7^2 + 10^2 – 2 \cdot 7 \cdot 10 \cdot \frac{\sqrt{2}}{2}
\]
\[
c^2 = 49 + 100 – 70\sqrt{2}
\]
\[
c^2 = 149 – 70\sqrt{2}
\]
Ukubala inani le-\( c \):
\[
c \cishe \sqrt{149 – 70\sqrt{2}} \cishe 5.97 \, \text{cm}
\]
Ngakho-ke, ubude bohlangothi lwesithathu lonxantathu bungaba ngu-5.97 cm.
Isiphetho
I-Trigonometry iyithuluzi eliwusizo kakhulu ekuxazululeni izinkinga ezahlukahlukene ezihilela onxantathu nama-engeli. Ngokuqonda kahle i-sine, i-cosine, i-tangent, kanye nemithetho ye-trigonometry, singaxazulula izinkinga eziningi ezisebenzayo. Lesi sihloko sixoxa ngezibonelo eziningana zokusebenzisa izilinganiso ze-trigonometric, esithemba ukuthi zizosiza abafundi ukuba baqonde kangcono.
