Mehlala ea Lipotso tse Buisanang ka Litekanyetso tse Tharo tsa Trigonometric
Trigonometry ke lekala la lipalo le ithutang kamano pakeng tsa bolelele le likhutlo ka likhutlotharo. E 'ngoe ea likhopolo tsa motheo ho trigonometry ke likarohano tsa trigonometric: sine (sin), cosine (cos), le tangent (tan). Sehlooho sena se tla akaretsa mathata a 'maloa a mehlala le puisano e felletseng ea likarohano tsa trigonometric ho nolofatsa kutloisiso ea hau.
1. Ho utloisisa Litekanyetso tse Tharo tsa Trigonometric
Sa pele, a re utloisiseng hore na ho boleloa eng ka sine, cosine le tangent.
– Sine (sepheo) sa sekhutlo ke karolelano ya bolelele ba lehlakore le fapaneng la sekhutlo le bolelele ba hypotenuse ya khutlotharo.
– Cosine (cos) ea sekhutlo ke karolelano ea bolelele ba lehlakore le haufi la sekhutlo le bolelele ba hypotenuse ea khutlotharo.
– Tangent (tan) ea sekhutlo ke karolelano ea bolelele ba lehlakore le fapaneng la sekhutlo le bolelele ba lehlakore le haufi. Tangent e ka boela ea hlalosoa e le quotient ea sine le cosine: tan(θ) = sin(θ) / cos(θ).
2. Lipotso tsa Mehlala le Puisano
Potso ea 1:
Fanoa ka khutlotharo e nepahetseng e nang le hypotenuse ea 10 cm le lehlakore le fapaneng la θ la 6 cm. Fumana boleng ba sin, cos, le tan ea angle θ.
Puisano:
Ho fumana boleng ba sin(θ), cos(θ), le tan(θ), re boetse re hloka ho tseba bolelele ba lehlakore le haufi. A re sebeliseng khopolo-taba ea Pythagorean ho fumana bolelele ba lehlakore le haufi.
Khopolo-taba ea Pythagorean:
\[ a^2 + b^2 = c^2 \]
moo c e leng hypotenuse, a ke lehlakore le fapaneng la sekhutlo, mme b ke lehlakore le haufi la sekhutlo.
E fanoe:
– Hypotenuse (c) = 10 cm
– Lehlakore le ka pele la sekhutlo θ (a) = 6 cm
Kahoo:
\[ a^2 + b^2 = c^2 \]
\[ 6^2 + b^2 = 10^2 \]
\[ 36 + b^2 = 100 \]
\[ b^2 = 64 \]
\[ b = \sqrt{64} \]
\[b = 8 \]
Kahoo, bolelele ba lehlakore (b) ke 8 cm.
Ka mor'a moo, re ka bala boleng ba sine, cosine, le tangent:
– Sin(θ) = Lehlakore le Hanyetsang / Hypotenuse
\[ \ sebe(θ) = \frac{6}{10} = 0.6 \]
– Cos(θ) = Lehlakore le ka thoko / Hypotenuse
\[ \cos(θ) = \frac{8}{10} = 0.8 \]
– Tan(θ) = Lehlakore le ka Pele / Lehlakore le ka Lehlakoreng
\[ \tan(θ) = \frac{6}{8} = 0.75 \]
Potso ea 2:
Ha ho fanoe ka khutlotharo e nepahetseng e bolelele ba lehlakore le fapaneng la sekhutlo sa α ke 5 cm mme bolelele ba lehlakore le haufi la sekhutlo sa α ke 12 cm. Fumana boleng ba sin, cos, le tan ea sekhutlo sa α.
Puisano:
Feela joalo ka potsong ea 1, ha re sebeliseng khopolo-taba ea Pythagorean ho fumana bolelele ba hypotenuse.
E fanoe:
– Lehlakore le ka pele la sekhutlo α (a) = 5 cm
– Lehlakore la sekhutlo α (b) = 12 cm
Sebelisa khopolo-taba ea Pythagorean:
\[ a^2 + b^2 = c^2 \]
\[ 5^2 + 12^2 = c^2 \]
\[ 25 + 144 = c^2 \]
\[ 169 = c^2 \]
\[ c = \sqrt{169} \]
\[c = 13 \]
Kahoo, bolelele ba hypotenuse (c) ke 13 cm.
Ka mor'a moo, re ka bala boleng ba sine, cosine, le tangent:
– Sin(α) = Lehlakore le Hanyetsang / Hypotenuse
\[ \sin(α) = \frac{5}{13} \]
– Cos(α) = Lehlakore le ka thoko / Hypotenuse
\[ \cos(α) = \frac{12}{13} \]
– Tan(α) = Lehlakore le ka Pele / Lehlakore
\[ \tan(α) = \frac{5}{12} \]
Potso ea 3:
Haeba ho tsebahala hore sin β = 0.6 le angle β li ho quadrant I, fumana boleng ba cos β le tan β.
Puisano:
Ho fanoe ka sebe β = 0.6
Rea tseba hore karolong ea I boleng ba cos β le bona bo nepahetse.
Sebelisa boitsebiso ba motheo ba trigonometric:
\[ \ sebe^2(β) + \cos^2(β) = 1 \]
\[ (0.6)^2 + \cos^2(β) = 1 \]
\[ 0.36 + \cos^2(β) = 1 \]
\[ \cos^2(β) = 1 – 0.36 \]
\[ \cos^2(β) = 0.64 \]
\[ \cos(β) = \sqrt{0.64} \]
\[ \cos(β) = 0.8 \]
Ka mor'a moo, re ka bala boleng ba tangent:
\[ \tan(β) = \frac{\sin(β)}{\cos(β)} \]
\[ \tan(β) = \frac{0.6}{0.8} \]
\[ \tan(β) = 0.75 \]
3. Kesimpulan
Khopolo ea trigonometric triad (sin, cos, tan) ke ea bohlokoa ebile ea bohlokoa ho utloisiseng trigonometry ka kakaretso. Ka ho utloisisa mokhoa oa ho fumana le ho bala boleng bona bo meraro mefuteng e fapaneng ea likhutlotharo, u ka rarolla mathata a mangata a fapaneng a trigonometry. Mathata a tšohliloeng ka holimo a lokela ho u thusa ho utloisisa mokhoa oa ho sebelisa likhopolo tsena maemong a fapaneng.
Kutloisiso e tiileng ea trigonometry e tla boela e u nolofalletse ho ithuta lihlooho tse tsoetseng pele haholoanyane lipalo le mahlale, tse kang calculus le fisiks. Se ke oa tsilatsila ho tsoela pele ho ikoetlisa le ho tebisa kutloisiso ea hau ea likhopolo tsena ho fihlela boemo bo phahameng ba boiphihlelo.
