Lipotso le Puisano tsa Mohlala oa Trigonometry
Trigonometry ke lekala la lipalo le ithutang kamano pakeng tsa likhutlo le bolelele ba mahlakoreng ka likhutlotharo. Ho utloisisa likhopolo tsa motheo tsa trigonometry ho bohlokoa lits'ebetsong tse fapaneng, ho tloha fisiks le boenjiniere ho ea ho bolepi ba linaleli le jeokrafi. Sehloohong sena, re tla hlalosa mehlala e 'maloa ea mathata le litlhaloso tse felletseng ho thusa kutloisiso.
Mohlala oa Potso ea 1: Ho Bala Mahlakore a Khutlotharo ho Sebelisa Mokhoa oa Sine
Potso:
Fana ka khutlotharo ABC, e nang le khutlo A = 30°, khutlo B = 45°, le lehlakore b = 10 cm. Bala bolelele ba lehlakore a.
Puisano:
Sebelisa molao oa libe:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Kenya boleng bo tsebahalang:
\[ \frac{a}{\sin 30°} = \frac{10}{\sin 45°} \]
Rea tseba hore:
\[ \sin 30° = \frac{1}{2} \]
\[ \sin 45° = \frac{\sqrt{2}}{2} \]
Jwale, kenya boleng bona sebakeng sa equation:
\[ \frac{a}{\frac{1}{2}} = \frac{10}{\frac{\sqrt{2}}{2}} \]
Nolofatsa equation:
\[ 2a = \frac{10 \cdot 2}{\sqrt{2}} \]
\[ 2a = \frac{20}{\sqrt{2}} \]
Beha mabaka a hore na ke bokae:
\[ 2a = \frac{20 \cdot \sqrt{2}}{2} \]
\[ 2a = 10\sqrt{2} \]
\[ a = 5\sqrt{2} \]
Kahoo, bolelele ba lehlakore a ke \( 5\sqrt{2} \) cm.
Mohlala oa Potso ea 2: Ho Bala Likhutlo ho Sebelisa Mokhoa oa Cosine
Potso:
Khutlotharo e na le mahlakore a = 7 cm, b = 10 cm, le c = 5 cm. Fumana boholo ba sekhutlo sa C.
Puisano:
Sebelisa molao oa li-cosine:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
Kenya boleng bo tsebahalang:
\[ 5^2 = 7^2 + 10^2 – 2 \cdot 7 \cdot 10 \cdot \cos C \]
Nolofatsa equation:
\[ 25 = 49 + 100 – 140 \ cos C \]
\[ 25 = 149 – 140 \ cos C \]
Suthela 149 ka lehlakoreng le letšehali:
\[ 25 – 149 = -140 \cos C \]
\[ -124 = -140 \cos C \]
\[ \cos C = \frac{124}{140} \]
\[ \cos C = \frac{62}{70} \]
\[ \cos C = \frac{31}{35} \]
Sebelisa khalkhuleita ho fumana \( \cos^{-1} \) (inverse cosine):
\[ C \hoo e ka bang \cos^{-1}\left(\frac{31}{35}\right) \]
\[ C \hoo e ka bang 25.84° \]
Kahoo, boholo ba angle C bo ka ba 25.84°.
Mohlala oa Potso ea 3: Ho Bala Bophahamo le Sebaka sa Khutlotharo
Potso:
Khutlotharo e na le mahlakore a mabedi a bolelele ba a = 6 cm le b = 8 cm ka sekhutlo se pakeng tsa ona, \(\theta\) = 60°. Bala bophahamo le sebaka sa khutlotharo.
Puisano:
1. Ho Bala Sebaka sa Khutlotharo:
Sebelisa foromo bakeng sa sebaka sa khutlotharo:
\[ \text{Area} = \frac{1}{2} ab \sin \theta \]
Kenya boleng bo tsebahalang:
\[ \text{Area} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \sin 60° \]
Rea tseba hore:
\[ \sin 60° = \frac{\sqrt{3}}{2} \]
Kahoo:
\[ \text{Area} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{Area} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \mongolo{Sebaka} = 24 \cdot \frac{\sqrt{3}}{2} \]
\[ \sebaka sa mongolo {Sebaka} = 12\sqrt{3} \]
Kahoo, sebaka sa khutlotharo ke \( 12\sqrt{3} \) cm².
2. Ho Bala Bophahamo ba Khutlotharo ho tloha Motheong a:
Ho bala bophahamo ba khutlotharo, bontsha bophahamo e le h mme o sebedise foromo ya sebaka:
\[ \text{Area} = \frac{1}{2} \times a \times h \]
\[ 12\sqrt{3} = \frac{1}{2} \makgetlo a 6 \makgetlo a h \]
\[ 12\sqrt{3} = 3h \]
\[ h = \frac{12\sqrt{3}}{3} \]
\[ h = 4\sqrt{3} \]
Kahoo, bophahamo ba khutlotharo ke \( 4\sqrt{3} \) cm.
Mohlala oa Potso ea 4: Ho Fumana Mahlakore a Khutlotharo e Letona
Potso:
Kgutlotharo e ka letsohong le letona, e nang le sekhutlo \(\theta\) = 30° mme lehlakore le bapileng la sekhutlo \(\theta\) e le 5 cm, fumana bolelele ba hypotenuse.
Puisano:
Sebelisa dikarolelano tsa trigonometric bakeng sa angle ya 30° ka khutlotharo e nepahetseng:
\[ \sin \theta = \frac{\text{front}}{\text{hypotenuse}} \]
\[ \sin 30° = \frac{obverse}{hypotenuse} = \frac{5}{\text{hypotenuse}} \]
Rea tseba hore:
\[ \sin 30° = \frac{1}{2} \]
Kahoo:
\[ \frac{1}{2} = \frac{5}{\text{hypotenuse}} \]
\[ \text{hypotenuse} = 10 \]
Kahoo, bolelele ba hypotenuse ke 10 cm.
Mohlala oa Potso ea 5: Ho Bala Likhutlo ka Mesebetsi ea Trigonometric
Potso:
Haeba \( \tan \theta = \frac{3}{4} \), bala boholo ba sekhutlo \(\theta\).
Puisano:
Ho sebelisa foromo ea tangent e fapaneng:
\[ \theta = \tan^{-1} \left(\frac{3}{4}\right) \]
Ka thuso ea khalkhuleita:
\[ \theta \hoo e ka bang 36.87° \]
Kahoo, boholo ba sekhutlo \(\theta\) bo ka ba 36.87°.
Qetello
Trigonometry ke khopolo e pharaletseng ea lipalo e nang le ts'ebeliso e pharaletseng mafapheng a fapaneng. Ho utloisisa mokhoa oa ho sebelisa melao ea sine, li-cosine le mesebetsi e meng ea motheo ho re lumella ho rarolla mathata a fapaneng a amanang le likhutlo tse tharo le likhutlo. Ka mehlala e kaholimo, ho tšeptjoa hore babali ba tla fumana kutloisiso e tebileng 'me ba khone ho e sebelisa maemong a fapaneng a hlokang kutloisiso ea trigonometry.
