Ajụjụ na Mkparịta ụka Ihe Nlereanya Trigonometry
Trigonometry bụ ngalaba mgbakọ na mwepụ nke na-amụ mmekọrịta dị n'etiti akụkụ na ogologo akụkụ na triangles. Ịghọta echiche ndị bụ isi nke trigonometry dị oke mkpa n'ọtụtụ ojiji, site na physics na injinia ruo na mbara igwe na ọdịdị ala. N'isiokwu a, anyị ga-akọwa ọtụtụ nsogbu atụ na nkọwa zuru oke iji nyere aka ịghọta.
Ajụjụ Ihe Nlereanya nke 1: Ịgbakọ Akụkụ nke Triangle Site na Iji Usoro Sine
Ajụjụ:
E nyere gị triangle ABC, nke nwere akụkụ A = 30°, akụkụ B = 45°, na akụkụ b = 10 cm. Gbakọọ ogologo akụkụ a.
Azịza:
Jiri iwu nke sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Tinye ụkpụrụ ndị a maara:
\[ \frac{a}{\sin 30°} = \frac{10}{\sin 45°} \]
Anyị maara na:
\[ \sin 30° = \frac{1}{2} \]
\[ \sin 45° = \frac{\sqrt{2}}{2} \]
Ugbu a, dochie ụkpụrụ ndị a n'ime nha nhata:
\[ \frac{a}{\frac{1}{2}} = \frac{10}{\frac{\sqrt{2}}{2}} \]
Mee ka usoro ahụ dị mfe:
\[ 2a = \frac{10 \cdot 2}{\sqrt{2}} \]
\[ 2a = \frac{20}{\sqrt{2}} \]
Mee ka ihe nkesa ahụ doo anya:
\[ 2a = \frac{20 \cdot \sqrt{2}}{2} \]
\[ 2a = 10\sqrt{2} \]
\[a = 5\sqrt{2} \]
Ya mere, ogologo akụkụ a bụ \(5\sqrt{2} \) cm.
Ihe atụ Ajụjụ nke 2: Ịgbakọ Akuku Site na Iji Usoro Cosine
Ajụjụ:
Triangle nwere akụkụ a = 7 cm, b = 10 cm, na c = 5 cm. Chọpụta nha akụkụ C.
Azịza:
Jiri iwu nke cosines:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
Tinye ụkpụrụ ndị a maara:
\[ 5^2 = 7^2 + 10^2 – 2 \cdot 7 \cdot 10 \cdot \cos C \]
Mee ka usoro ahụ dị mfe:
\[ 25 = 49 + 100 - 140 \ cos C \]
\[ 25 = 149 – 140 \cos C \]
Bugharịa 149 gaa n'akụkụ aka ekpe:
\[ 25 - 149 = -140 \ cos C \]
\[ -124 = -140 \cos C \]
\[ \cos C = \frac{124}{140} \]
\[ \cos C = \frac{62}{70} \]
\[ \cos C = \frac{31}{35} \]
Jiri ihe mgbakọ na mwepụ chọta \( \cos^{-1} \) (cosine dị iche):
\[ C \approx \cos^{-1}\left(\frac{31}{35}\right) \]
\[ C \ihe dị ka 25.84° \]
Ya mere, nha nke akụkụ C dị ihe dịka 25.84°.
Ajụjụ Ihe Nlereanya nke 3: Ịgbakọ Ogologo na Mpaghara nke Triangle
Ajụjụ:
Triangle nwere akụkụ abụọ nke ogologo a = 6 cm na b = 8 cm yana akụkụ dị n'etiti ha, \(\theta\) = 60 °. Gbakọọ ịdị elu na mpaghara nke triangle ahụ.
Azịza:
1. Ịgbakọ Mpaghara nke Triangle:
Jiri usoro maka mpaghara triangle:
\[ \text{Area} = \frac{1}{2} ab \sin \theta \]
Tinye ụkpụrụ ndị a maara:
\[ \text{Mpaghara} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \sin 60° \]
Anyị maara na:
\[ \sin 60° = \frac{\sqrt{3}}{2} \]
Ya mere:
\[ \text{Mpaghara} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{Mpaghara} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{Mpaghara} = 24 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{Mpaghara} = 12\sqrt{3} \]
Ya mere, ógbè triangle ahụ bụ cm².
2. Ịgbakọ Ogologo nke Triangle site na Ntọala a:
Iji gbakọọ elu nke triangle, gosi elu dị ka h wee jiri usoro mpaghara:
\[ \text{Area} = \frac{1}{2} \times a \times h \]
\[ 12\sqrt{3} = \frac{1}{2} \ugboro isii \ugboro h \]
\[ 12\sqrt{3} = awa 3 \]
\[ h = \frac{12\sqrt{3}}{3} \]
\[ h = 4\sqrt{3} \]
Ya mere, elu nke triangle ahụ bụ \( 4 \sqrt{3} \) cm.
Ihe atụ Ajụjụ nke 4: Ịchọpụta Akụkụ nke Triangle Aka Nri
Ajụjụ:
N'ime triangle aka nri, ya na akụkụ \(\theta\) = 30° na akụkụ parallel nke akụkụ \(\theta\) bụ 5 cm, chọpụta ogologo nke hypotenuse.
Azịza:
Jiri oke trigonometric maka akụkụ 30° na triangle aka nri:
\[ \sin \theta = \frac{\text{front}}{\text{hypotenuse}} \]
\[ \sin 30° = \frac{obverse}{hypotenuse} = \frac{5}{\text{hypotenuse}} \]
Anyị maara na:
\[ \sin 30° = \frac{1}{2} \]
Ya mere:
\[ \frac{1}{2} = \frac{5}{\text{hypotenuse}} \]
\[ \text{hypotenuse} = 10 \]
Ya mere, ogologo nke hypotenuse bụ 10 cm.
Ajụjụ Ihe atụ nke 5: Ịgbakọ Angles site na iji ọrụ Trigonometric
Ajụjụ:
Ọ bụrụ na \( \tan \theta = \frac{3}{4} \), gbakọọ nha nke akụkụ \(\theta\).
Azịza:
Site na iji usoro inverse tangent:
\[ \theta = \tan^{-1} \left(\frac{3}{4}\right) \]
Site n'enyemaka nke ihe mgbakọ na mwepụ:
\[ \theta \ihe dịka 36.87° \]
Ya mere, nha nke akụkụ \(\theta\) dị ihe dị ka 36.87°.
Mmechi
Trigonometry bụ echiche mgbakọ na mwepụ sara mbara nke nwere ọtụtụ ojiji n'ọtụtụ ngalaba dị iche iche. Ịghọta otu esi eji iwu sines, cosines, na ọrụ ndị ọzọ bụ isi na-enye anyị ohere idozi ọtụtụ nsogbu metụtara triangles na akụkụ. Site na ihe atụ ndị dị n'elu, a na-atụ anya na ndị na-agụ akwụkwọ ga-enweta nghọta miri emi ma nwee ike itinye ya n'ọrụ n'ọnọdụ dị iche iche chọrọ nghọta nke trigonometry.
