Mienzaniso yeMibvunzo Inokurukura Mabasa eTrigonometric
Mabasa eTrigonometric chikamu chakakosha chemasvomhu, anowanzoonekwa munzvimbo dzakasiyana dzesainzi, kusanganisira fizikisi, mainjiniya, uye sainzi yemakombiyuta. Muchinyorwa chino, tichakurukura matambudziko akati wandei uye kupa hurukuro yakadzama yemabasa etrigonometric. Nekunzwisisa mienzaniso iyi, vaverengi vanotarisira kusimbisa kunzwisisa kwavo nekugona kwavo kugadzirisa matambudziko ane chekuita nemabasa etrigonometric.
Nhanganyaya kuMabasa eTrigonometric
Mashandiro akajairika etrigonometric ndeanoti sine (sin), cosine (cos), uye tangent (tan). Mashandiro matatu aya anoita basa rakakosha muhukama huripo pakati pemakona nehurefu mumatriangles ekurudyi, pamwe chete nemafungu nekudengenyeka.
Mafomura Ekutanga:
1. Sine (chivi)
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
2. Kosine (cos)
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
3. Kutsvuka (kusviba)
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Kuzivikanwa kweTrigonometric
- Pythagoras:
\[
\chivi^2(\theta) + \cos^2(\theta) = 1
\]
- Kuenzanisa tangent ne sine ne cosine:
\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\]
– Kuwedzera kuzivikanwa:
\[
\chivi(2\theta) = 2\chivi(\theta)\cos(\theta)
\]
\[
\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta)
\]
Ngatitarisei mimwe mienzaniso yemibvunzo uye hurukuro yakadzama.
Muenzaniso Mubvunzo 1: Kuverenga Kukosha kweMabasa eTrigonometric paAngle Yakati
Mubvunzo:
Verenga kukosha kwe sin(30°), cos(45°), uye tan(60°).
Kukurukurirana:
Zvichienderana netafura yehukuru hwe trigonometric, tine:
– \(\chivi(30°) = \frac{1}{2} = 0.5\)
– \(\cos(45°) = \frac{\sqrt{2}}{2} \inenge 0.707\)
– \(\tan(60°) = \sqrt{3} \inenge 1.732\)
Zvinhu zvitatu zviri pamusoro apa zviyero zve trigonometric zvinowanzo shandiswa, uye zvakanaka kuzvibata nemusoro nekuti zvinowanzoonekwa mumibvunzo.
Muenzaniso Mubvunzo 2: Kuverenga Makona Uchishandisa Mashandiro eInverse Trigonometric
Mubvunzo:
Kana \(\sin(\theta) = 0.5\), sarudza kukosha kwe \(\theta\).
Kukurukurirana:
Kuti tiwane kukosha kwe \(\theta\), tinofanira kushandisa basa re "inverse" re "sine", kureva \(\arcsin\) kana \(\sin^{-1}\).
\[
\theta = \chivi^{-1}(0.5)
\]
Muchikamu che [0°, 360°], kukosha kwakafanana kwe \(\theta\) ndekwekuti:
\[
\theta = 30° \text{ uye } 150°
\]
nekuti \(\sin(30°) = 0.5\) uye \(\sin(150°) = 0.5\). Saka, makona maviri anogutsa ndiwo 30° ne150°.
Muenzaniso Mubvunzo 3: Kushandisa Trigonometric Identities
Mubvunzo:
Ratidza hunhu hwetrigonometric
\[
\chivi^2(\theta) + \cos^2(\theta) = 1.
\]
Kukurukurirana:
Kuzivikanwa uku kunobva mudzidziso yePythagorean iri mumatriangles ekurudyi. Ngatitii pane triangle yekurudyi ine kona \(\theta\), divi rakatarisana \(a\), divi riri pedyo \(b\), uye hypotenuse \(c\). Zvadaro,
\[
a^2 + b^2 = c^2.
\]
Kana tikakamura mativi ese ari maviri ne \(c^2\), tinowana:
\[
\left(\frac{a}{c}\right)^2 + \left(\frac{b}{c}\right)^2 = 1.
\]
Nokuti
\[
\sin(\theta) = \frac{a}{c} \quad \text{and} \quad \cos(\theta) = \frac{b}{c},
\]
saka,
\[
\chivi^2(\theta) + \cos^2(\theta) = 1.
\]
Ndiyo nzira yatinoratidza nayo hunhu uhwu.
Muenzaniso Mubvunzo 4: Kushandisa Mabasa eTrigonometric mukugadzirisa maTriangles
Mubvunzo:
Tine mativi matatu eABC ane kona yeA 45°, kona yeB 60°, uye mativi eAB ane kureba kwe10 cm. Tsvaga hurefu hwemativi eAC neBC.
Kukurukurirana:
Shandisa mutemo wesine kuti uwane hurefu hwemativi AC neBC.
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]
Kutanga, tinowana kona C:
\[
C = 180° – A – B = 180° – 45° – 60° = 75°.
\]
Ne AB = 10 cm, \(A = 45°\), uye \(B = 60°\), tinogona kushandisa mutemo we sine:
\[
\frac{AC}{\sin(60°)} = \frac{10}{\sin(75°)}.
\]
\[
AC = \frac{10 \chivi(60°)}{\sin(75°)}.
\]
\[
AC = \frac{10 \times \frac{\sqrt{3}}{2}}{\sin(75°)} = \frac{10 \times \frac{\sqrt{3}}{2}}{\cos(15°)}.
\]
Tinoziva kuti \(\cos(15°) = \cos(45° – 30°) = \cos 45° \cos 30° + \sin 45° \sin 30° = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{2}\).
\[
\cos(15°) = \frac{\sqrt{6} + \sqrt{2}}{4}.
\]
Kuti:
\[
AC = \frac{10 \times \frac{\sqrt{3}}{2}}{\frac{\sqrt{6} + \sqrt{2}}{4}} = \frac{10 \times 2\sqrt{3}}{\sqrt{6} + \sqrt{2}} \approx 10.39 \text{cm}.
\]
Nenzira yakafanana, tinogona kuwana BC:
\[
\frac{BC}{\sin(45°)} = \frac{10}{\sin(75°)}.
\]
\[
BC = \frac{10 \sin(45°)}{\sin(75°)} \approx 8.66 \text{ cm}.
\]
Mukupedzisa kwechinyorwa chino, takurukura nezvematambudziko akati wandei emuenzaniso uye hurukuro dzawo maererano nemabasa etrigonometric. Nekudzidzira kwakasimba uye kunzwisisa kwakakwana kwemafomula ekutanga, hunhu hwetrigonometric, uye mashandisirwo awo mumatriangles, vaverengi vanotarisirwa kunyatsogona izvi. Mabasa etrigonometric zvishandiso zvakakosha, kwete mumasvomhu chete asiwo muzvidzidzo zvakasiyana-siyana zvinotsamira pakuongorora makona nehurefu. Tinovimba kuti chinyorwa ichi chave chirevo chinobatsira vaverengi.
