Mehlala ea Lipotso tse Buisanang ka Litekanyetso tsa Trigonometric
Trigonometry ke lekala la lipalo le ithutang likamano pakeng tsa mahlakore le likhutlo tsa khutlotharo. Karolo e 'ngoe ea bohlokoa ea trigonometry ke ho utloisisa likarohano tsa trigonometric, ho kenyeletsoa sine (sin), cosine (cos), le tangent (tan). Sehlooho sena se tla tšohla mehlala e 'maloa ea likarohano tsa trigonometric le litlhaloso tsa tsona tse qaqileng ho nolofatsa kutloisiso e tebileng ea likhopolo tsa motheo tsa trigonometric.
Mohlala oa Potso ea 1: Ho Bala Litekanyetso tsa Sebe, Cos, le Tan
Potso:
Seithuti se kopuoa ho fumana boleng ba sin, cos, le tan ea angle \( \theta \) ka khutlotharo e nepahetseng, haeba bolelele ba lehlakore le fapaneng (angle e fapaneng \(\theta \)) e le 3 cm, bolelele ba lehlakore le haufi (botlaase) ke 4 cm, 'me bolelele ba hypotenuse (hypotenuse) e le 5 cm.
Puisano:
Mohato oa pele ke ho khetholla lehlakore ka leng le amanang le sekhutlo se fanoeng. Joalokaha ho tsejoa, ka khutlotharo e nepahetseng e nang le mahlakore a triple ea khale ea Pythagorean (3, 4, 5), re ka sebelisa mekhoa ea motheo ea trigonometric ka ho toba.
– Sine (sebe) ke karolelano pakeng tsa bolelele ba lehlakore le fapaneng le hypotenuse.
\[
\sin(\theta) = \frac{\text{obverse side}}{\text{hypotenuse}} = \frac{3}{5}
\]
– Cosine (cos) ke karolelano pakeng tsa bolelele ba lehlakore le haufi le hypotenuse.
\[
\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{4}{5}
\]
– Tangent (botsho) ke karolelano pakeng tsa bolelele ba lehlakore le ka pele le lehlakore.
\[
\tan(\theta) = \frac{\text{front side}}{\text{side side}} = \frac{3}{4}
\]
Kahoo, boleng ba dikarolelano tsa trigonometric bakeng sa angle \(\theta\) ke:
\[
\sin(\theta) = \frac{3}{5}, \quad \cos(\theta) = \frac{4}{5}, \quad \tan(\theta) = \frac{3}{4}
\]
Mohlala oa Potso ea 2: Ho Fumana Mahlakore a Khutlotharo ho Sebelisa Litekanyetso tsa Trigonometric
Potso:
Fana ka khutlotharo e nepahetseng e nang le sekhutlo \( \alpha = 30^\circ \). Haeba bolelele ba lehlakore le fapaneng la sekhutlo \( \alpha \) e le 4 cm, fumana bolelele ba lehlakore le leng.
Puisano:
Ho sebelisa boleng ba karolelano ea trigonometric bakeng sa angle \( 30^\circ \):
– \(\sin(30^\circ) = \frac{1}{2} \)
\[
\sin(30^\circ) = \frac{\text{front side}}{\text{hypotenuse}} = \frac{4}{\text{hypotenuse}}
\]
\[
\text{Hypotenuse} = \frac{4}{\sin(30^\circ)} = \frac{4}{\frac{1}{2}} = 8 \text{ cm}
\]
– \(\cos(30^\circ) = \frac{\sqrt{3}}{2} \)
\[
\cos(30^\circ) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\text{adjacent side}}{8}
\]
\[
\text{Side side} = 8 \cos(30^\circ) = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3} \text{cm}
\]
Kahoo, bolelele ba lehlakore le leng ke \(4\sqrt{3}\) cm mme hypotenuse ke 8 cm.
Mohlala oa 3: Ho Sebelisa Trigonometry ho Cartesian Coordinates
Potso:
Fumana boleng ba sin, cos, le tan bakeng sa ntlha \( P(3, 4) \) ho li-coordinate tsa Cartesian haeba ntlha e etsa sekhutlo \(\theta\) ka x-axis e ntle ho quadrant I.
Puisano:
Taba ea pele, bala sebaka sa ntlha \(P(3, 4)\) ho tloha tšimolohong (O), e leng hypotenuse ea khutlotharo e entsoeng.
– Hypotenuse (\(r\)) e ka balwa ho sebediswa theorem ya Pythagorean:
\[
r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Ebe, trigonometry ea ntlha P e ka baloa ka tsela e latelang:
– Sine (sin) ke karolelano pakeng tsa ordinate (y) le hypotenuse (r):
\[
\sin(\theta) = \frac{y}{r} = \frac{4}{5}
\]
– Cosine (cos) ke karolelano pakeng tsa abscissa (x) le hypotenuse (r):
\[
\cos(\theta) = \frac{x}{r} = \frac{3}{5}
\]
– Tangent (tan) ke karolelano pakeng tsa ordinate (y) le abscissa (x):
\[
\tan(\theta) = \frac{y}{x} = \frac{4}{3}
\]
Kahoo, boleng ba trigonometric bakeng sa ntlha P ke:
\[
\sin(\theta) = \frac{4}{5}, \quad \cos(\theta) = \frac{3}{5}, \quad \tan(\theta) = \frac{4}{3}
\]
Mohlala oa Potso ea 4: Ho Sebelisa Boitsebiso ba Trigonometric
Potso:
Haeba \( \sin(\theta) = \frac{3}{5} \), fumana boleng ba \( \cos(\theta) \) o sebedisa boitsebiso ba trigonometric \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
Puisano:
Re qala ka boitsebiso ba motheo ba trigonometric:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
Ha ho fanoe ka \( \sin(\theta) = \frac{3}{5} \), ebe:
\[
\sin^2(\theta) = \left(\frac{3}{5}\right)^2 = \frac{9}{25}
\]
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
\[
\frac{9}{25} + \cos^2(\theta) = 1
\]
\[
\cos^2(\theta) = 1 – \frac{9}{25} = \frac{25}{25} – \frac{9}{25} = \frac{16}{25}
\]
\[
\cos(\theta) = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}
\]
Boleng ba \( \cos(\theta) \) e ka ba bo botle kapa bo bobe ho latela quadrant eo angle \(\theta\) e leng ho yona. Empa bakeng sa bothata bona ntle le tlhaloso e hlakileng ya quadrant, re sebedisa \(\cos(\theta) = \frac{4}{5}\) bakeng sa quadrant I kapa \( -\frac{4}{5}\) bakeng sa quadrants II, III, kapa IV.
Qetello
Ho utloisisa dikarolelano tsa trigonometric ho bohlokoa bakeng sa ho rarolla mathata a fapaneng a kenyeletsang dikhutlo le bolelele dikhutlotharong. Ka ho itloaetsa ho rarolla mathata a kang a ka hodimo, o ka ntlafatsa bokgoni ba hao ba ho bala le ho utlwisisa dikamano pakeng tsa mahlakore le dikhutlo dikhutlotharong. Trigonometry ha e sebediswe feela dipalong tse hlwekileng empa hape le mafapheng a fapaneng a saense jwalo ka fisiks, boenjiniere le bolepi ba dinaledi. Ho thabile ho itloaetsa!
