Mehlala ea Lipotso Tse Buisanang ka Tšebeliso ea Litekanyetso tsa Trigonometric
Trigonometry ke lekala la lipalo le ithutang likamano pakeng tsa mahlakore le likhutlo tsa likhutlotharo. Ho utloisisa trigonometry ho bohlokoa hobane e sebelisoa khafetsa mafapheng a fapaneng, ho kenyeletsoa meralo ea meaho, boenjiniere, bolepi ba linaleli, esita le ho ngola ka mokhoa oa ho ngola. Sengoloa sena se tla tšohla mathata a 'maloa a mehlala le ho a tšohla moelelong oa ho sebelisa likarohano tsa trigonometry.
Mehopolo ea Motheo ea Trigonometry
Pele re kena mathateng a mohlala, ha re hlahlobeng mehopolo e meng ea motheo ho trigonometry. Khutlotharo e ka letsohong le letona, ho na le mesebetsi e meraro e meholo ea trigonometry e atisang ho sebelisoa: sine, cosine, le tangent.
– Sine (sekoli) sa sekhutlo ke karolelano ea bolelele ba lehlakore le shebaneng le sekhutlo le bolelele ba hypotenuse.
\[
\sin \theta = \frac{\text{obverse side}}{\text{hypotenuse}}
\]
– Cosine (cos) ea sekhutlo ke karolelano ea bolelele ba lehlakore le haufi le sekhutlo le bolelele ba hypotenuse.
\[
\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}
\]
– Tangent (botsho) ba sekhutlo ke karolelano ya bolelele ba lehlakore le shebaneng le sekhutlo le bolelele ba lehlakore le haufi le sekhutlo.
\[
\tan \theta = \frac{\text{front side}}{\text{side}}
\]
Mohlala oa Potso ea 1: Ho Bala Bophahamo ba Tora
Potso: Mohlokomeli o eme limithara tse 50 ho tloha toreng mme o lekanya sekhutlo sa bophahamo ba ntlha ya tora ho ba di-degree tse 30. Fumana bophahamo ba tora.
Tharollo: Ho rarolla bothata bona, re ka sebelisa mosebetsi oa tangent ho trigonometry. Kaha re tseba sekhutlo sa bophahamo le sebaka se bataletseng ho tloha ho moshebelli ho ea toreng, re ka ngola:
\[
\tan 30^\circ = \frac{\text{tower height}}{\text{horizontal distance}}
\]
Kenya mekhoa e tsebahalang sebakeng sa eona:
\[
\tan 30^\circ = \frac{h}{50}
\]
Hoa tsebahala hore \(\tan 30^\circ = \frac{1}{\sqrt{3}}\), e le hore:
\[
\frac{1}{\sqrt{3}} = \frac{h}{50}
\]
Ebe bophahamo ba tora, \(h\), bo ka fumanoa ka ho atisa mahlakore ka bobeli a equation ka 50:
\[
h = 50 \cdot \frac{1}{\sqrt{3}} = \frac{50}{\sqrt{3}} \hoo e ka bang 28.87 \, \text{meter}
\]
Bophahamo ba tora ke limithara tse ka bang 28.87.
Mohlala oa Potso ea 2: Ho Fumana Sebaka U Sebelisa Cosine
Potso: Sekepe se leba 10 km ka bochabela, ebe se fetola tsela ka di-degree tse 60 ka leboya mme se tsamaya ka di-km tse 15. Fumana sebaka ho tloha moo se qalang teng ho ya sekepeng.
Puisano: Ho rarolla bothata bona, re ka sebelisa molao oa cosine ho trigonometry. Haeba re rala leeto la sekepe ka sistimi ea coordinate, re fumana khutlotharo e nang le mahlakore a 10 km le 15 km, le sekhutlo sa likhato tse 60. Re ka sebelisa molao oa cosine ho fumana sebaka se pakeng tsa moo sekepe se qalang teng le sebaka sa sona sa ho qetela.
\[
c^2 = a^2 + b^2 – 2ab \cos C
\]
Moo:
– \( a = 10 \)
– \( b = 15 \)
– \( C = 60^\circ \)
Kenya mekhoa e tsebahalang sebakeng sa eona:
\[
c^2 = 10^2 + 15^2 – 2 \cdot 10 \cdot 15 \cdot \cos 60^\circ
\]
Re tseba hore \(\cos 60^\circ = 0.5\), ebe:
\[
c^2 = 100 + 225 – 2 \cdot 10 \cdot 15 \cdot 0.5
\]
\[
c^2 = 100 + 225 – 150
\]
\[
c^2 = 175
\]
\[
c = \sqrt{175} \hoo e ka bang 13.23 \, \mongolo{km}
\]
Kahoo sebaka ho tloha moo sekepe se qalang teng ho ea sekepeng ke hoo e ka bang 13.23 km.
Mohlala oa Potso ea 3: Ho Sebelisa Sine ho Fumana Mahlakore a Khutlotharo
Potso: Khutlotharo, mahlakore a mabedi ke 7 cm le 10 cm, mme sekhutlo pakeng tsa ona ke 45 degrees. Bala bolelele ba lehlakore la boraro la khutlotharo.
Puisano: Re ka sebelisa molao oa li-sine ho rarolla bothata bona. Molaong oa li-sine, bakeng sa khutlotharo e nang le mahlakore \(a\), \(b\), le \(c\) le sekhutlo \(C\) pakeng tsa mahlakore \(a\) le \(b\):
\[
\frac{c}{\sin C} = \frac{a}{\sin A} = \frac{b}{\sin B}
\]
Leha ho le jwalo, tabeng ena, re ka sebedisa molao wa di-cosine ka ho toba ho nolofatsa dintho. Molao wa di-cosine o re:
\[
c^2 = a^2 + b^2 – 2ab \cos C
\]
Moo:
– \( a = 7 \)
– \( b = 10 \)
– \( C = 45^\circ \)
\(\cos 45^\circ = \frac{\sqrt{2}}{2}\), e le hore:
\[
c^2 = 7^2 + 10^2 – 2 \cdot 7 \cdot 10 \cdot \frac{\sqrt{2}}{2}
\]
\[
c^2 = 49 + 100 – 70\sqrt{2}
\]
\[
c^2 = 149 – 70\sqrt{2}
\]
Ho bala boleng ba \( c \):
\[
c \hoo e ka bang \sqrt{149 – 70\sqrt{2}} \hoo e ka bang 5.97 \, \text{cm}
\]
Kahoo, bolelele ba lehlakore la boraro la khutlotharo ke hoo e ka bang 5.97 cm.
Qetello
Trigonometry ke sesebelisoa se thusang haholo bakeng sa ho rarolla mathata a fapaneng a amanang le likhutlo tse tharo le likhutlo. Ka kutloisiso e ntle ea sine, cosine, tangent, le melao ea trigonometry, re ka rarolla mathata a mangata a sebetsang. Sengoloa sena se tšohla mehlala e 'maloa ea ho sebelisa likarohano tsa trigonometry, tseo re tšepang hore li tla thusa babali ho ntlafatsa kutloisiso ea bona.
