Mienzaniso yeMibvunzo yeKukurukura nezveTrigonometric Ratios
Trigonometry ibazi remasvomhu rinobata nehukama huripo pakati pehurefu nemakona mumatriangles. Trigonometric ratios inoita basa guru muzvikamu zvakasiyana-siyana zvakaita seinjiniya, fizikisi, nyeredzi, uye kunyangwe hupenyu hwezuva nezuva. Chinyorwa chino chichaongorora mienzaniso yakati wandei yezvinetso nemhinduro dzazvo kuti zvikubatsire kunzwisisa pfungwa huru dzetrigonometric ratios.
Zviyero zveTrigonometric muRight Triangles
Ngatitangei nekunzwisisa zviyero zvekutanga mutriangle yekurudyi. Zviyero izvi zvinozivikanwa sesine (sin), cosine (cos), uye tangent (tan). Mutriangle yekurudyi, kune mativi matatu akakosha atinofanira kuziva:
1. Rutivi Rwakatarisana: Rutivi rwakatarisana zvakananga nekona yatiri kufunga nezvayo.
2. Rutivi Rwakatenderedza: Rutivi rwuri pedyo nekona yatiri kufunga nezvayo.
3. Hypotenuse: Rutivi rurefu pane rumwe rutivi rwekona yekurudyi rwakatarisana nekona yekurudyi.
Muenzaniso Mubvunzo 1
Tichipa bhora rerudyi rine kona θ, uko divi rakapesana nekona θ rine urefu hwemayuniti matatu, divi riri pedyo rine hurefu hwemayuniti mana, uye hypotenuse ine hurefu hwemayuniti mashanu. Tsvaga kukosha kwe sin, cos, uye tan yekona θ.
Mhinduro:
Kuti tiwane sine, cosine, uye tangent values ye angle θ, tinoshandisa basic trigonometric ratio formulas:
– Sine (chivi) θ = Front Side / Hypotenuse
\[ \sin θ = \frac{3}{5} \]
– Cosine (cos) θ = Rutivi rwekumativi / Hypotenuse
\[ \cos θ = \frac{4}{5} \]
– Tangent (tan) θ = Rutivi rweKumberi / Rutivi
\[ \tan θ = \frac{3}{4} \]
Hukama hweTrigonometric hweMamwe Makona
Zviyero zveTrigonometric zvinogonawo kushandiswa kune mamwe maangles mumhando dzakasiyana dzetriangles, kusanganisira triangles dzakaenzana uye triangles dzakajairwa. Zvakakosha kunzwisisa mitemo yekutanga senge Sine Rule neCosine Rule, iyo inoshanda kune triangles dzisiri right triangles.
Muenzaniso Mubvunzo 2
Tichipa katatu ABC ine mativi a = 7 cm, b = 24 cm, uye kona C = 90°. Tsvaga kureba kwerutivi c uye kukosha kwemakona A naB.
Mhinduro:
Sezvo kona C iri kona yekurudyi, tinogona kushandisa Pythagorean Theorem kuti tiwane hurefu hwerutivi c (hypotenuse):
\[ c = \sqrt{a^2 + b^2} \]
\[ c = \sqrt{7^2 + 24^2} \]
\[ c = \sqrt{49 + 576} \]
\[ c = \sqrt{625} \]
\[ c = 25 \, \chinyorwa{cm} \]
Tevere, kuti tione kukosha kwema angles A na B, tinogona kushandisa basic trigonometric ratios.
Pakuyera kona A, tinoshandisa cosine:
\[ \cos A = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{24}{25} \]
\[ A = \cos^{-1} \left(\frac{24}{25}\right) \]
Kune angle B, tinoshandisa sine:
\[ \sin B = \frac{\text{obverse side}}{\text{hypotenuse}} = \frac{24}{25} \]
\[ B = \chivi^{-1} \kuruboshwe(\frac{24}{25}\kurudyi) \]
Nekuti mutriangle yekurudyi huwandu hwemakona asiri ekurudyi hunofanira kunge huri 90°:
\[ A + B = 90° \]
\[ A = 90° – B \]
Mutemo weSine
Mutemo we sine unotsanangurwa seizvi:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Muenzaniso Mubvunzo 3
Tichipa mativi matatu ane divi a = 8 cm, divi b = 15 cm, uye angle C = 60°. Tsvaga angle A nehurefu hwedivi c uchishandisa Rule of Sines.
Mhinduro:
Kutanga, shandisa Mutemo weSines kuti uwane imwe yemakona mamwe (semuenzaniso, kona A):
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
\[ \frac{8}{\sin A} = \frac{c}{\sin 60°} \]
Tinofanira kutanga taverenga kukosha kwa c. Sezvo kona C iri 60°, tinogona kushandisa Mutemo weCosine:
\[ c^2 = a^2 + b^2 – 2ab \cos(C) \]
\[ c^2 = 8^2 + 15^2 – 2 \cdot 8 \cdot 15 \cdot \cos(60°) \]
\[ c^2 = 64 + 225 – 240 \cdot 0.5 \]
\[ c^2 = 289 – 120 \]
\[ c^2 = 169 \]
\[ c = 13 \, \chinyorwa{cm} \]
Zvino, tinogona kuwana kona A:
\[ \frac{8}{\sin A} = \frac{13}{\sin 60°} \]
Sezvo \(\sin 60° = \sqrt{3}/2\):
\[ \frac{8}{\sin A} = \frac{13}{\sqrt{3}/2} \]
\[ \frac{8}{\sin A} = \frac{26}{\sqrt{3}} \]
\[ 8 \cdot \sqrt{3} = 26 \sin A \]
\[ \sin A = \frac{8 \sqrt{3}}{26} \]
\[ \sin A = \frac{4 \sqrt{3}}{13} \]
Pakupedzisira, kushandisa inverse sine:
\[ A = \sin^{-1}\left(\frac{4 \sqrt{3}}{13}\right) \]
Mutemo weCosine
Mutemo wecosine unoti:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
Muenzaniso Mubvunzo 4
Tichipa mativi matatu ane divi a = 9 cm, divi b = 12 cm, uye divi c = 15 cm. Tsvaga kona C.
Mhinduro:
Kuti tiwane angle C, tinoshandisa Cosine Rule:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
\[ 15^2 = 9^2 + 12^2 – 2 \cdot 9 \cdot 12 \cdot \cos C \]
\[ 225 = 81 + 144 – 216 \cos C \]
\[225 = 225 – 216 \cos C \]
\[ 0 = -216 \cos C \]
\[ \cos C = 0 \]
Kubva pano tinoziva kuti kona C i90° nekuti 90° = 0.
Aya ndiwo mamwe emibvunzo yekukubatsira kunzwisisa trigonometric ratios. Kuramba uchidzidzira nematambudziko akasiyana-siyana kuchabatsira zvikuru kusimbisa kunzwisisa kwako pfungwa idzi. Tinovimba kuti chinyorwa chino chave chichibatsira mukudzidza kwako!
