Ajụjụ Ihe Nlereanya Na-atụle Oke Trigonometric
Trigonometry bụ ngalaba mgbakọ na mwepụ nke na-amụ mmekọrịta dị n'etiti akụkụ na akụkụ nke triangle. Otu akụkụ dị mkpa nke trigonometry bụ ịghọta oke trigonometric, gụnyere sine (sin), cosine (cos), na tangent (tan). Isiokwu a ga-atụle ọtụtụ ihe atụ nke oke trigonometric na nkọwa zuru ezu ha iji mee ka nghọta zuru oke nke echiche trigonometric bụ isi dị mfe.
Ajụjụ Ihe Nlereanya nke 1: Ịgbakọ Uru nke Mmehie, Cos, na Tan
Ajụjụ:
A na-agwa nwa akwụkwọ ka ọ chọta ụkpụrụ nke sin, cos, na tan nke akụkụ \(\theta \) na triangle aka nri, ọ bụrụ na ogologo nke akụkụ nke ọzọ (nkuku dị iche \(\theta \)) bụ 3 cm, ogologo nke akụkụ dị n'akụkụ (isi) bụ 4 cm, na ogologo nke hypotenuse (hypotenuse) bụ 5 cm.
Azịza:
Nzọụkwụ mbụ bụ ịchọpụta akụkụ ọ bụla metụtara akụkụ enyere. Dịka a maara, n'ime triangle aka nri nwere akụkụ nke atọ Pythagorean (3, 4, 5), anyị nwere ike iji usoro trigonometric bụ isi ozugbo.
– Sine (mmehie) bụ oke dị n'etiti ogologo nke akụkụ nke ọzọ na hypotenuse.
\[
\sin(\theta) = \frac{\text{obverse side}}{\text{hypotenuse}} = \frac{3}{5}
\]
– Cosine (cos) bụ oke dị n'etiti ogologo akụkụ dị n'akụkụ na hypotenuse.
\[
\cos(\theta) = \frac{\text{akụkụ dị n'akụkụ}}{\text{hypotenuse}} = \frac{4}{5}
\]
– Tangent (tan) bụ oke dị n'etiti ogologo nke akụkụ ihu na akụkụ.
\[
\tan(\theta) = \frac{\text{front side}}{\text{side side}} = \frac{3}{4}
\]
Ya mere, uru nke oke trigonometric maka akụkụ \(\theta\) bụ:
\[
\sin(\theta) = \frac{3}{5}, \quad \cos(\theta) = \frac{4}{5}, \quad \tan(\theta) = \frac{3}{4}
\]
Ihe atụ Ajụjụ nke 2: Ịchọta Akụkụ nke Triangle Site na Iji Usoro Trigonometric
Ajụjụ:
E nyere triangle aka nri nwere akụkụ \( \alpha = 30^\cir \). Ọ bụrụ na ogologo akụkụ nke ọzọ nke akụkụ \( \alpha \) bụ 4 cm, chọpụta ogologo akụkụ nke ọzọ.
Azịza:
Iji ụkpụrụ nke trigonometric ratio maka akụkụ \( 30^\cir \):
– \(\sin(30^\circ) = \frac{1}{2} \)
\[
\sin(30^\circ) = \frac{\text{front side}}{\text{hypotenuse}} = \frac{4}{\text{hypotenuse}}
\]
\[
\text{Hypotenuse} = \frac{4}{\sin(30^\circ)} = \frac{4}{\frac{1}{2}} = 8 \text{ cm}
\]
– \(\cos(30^\circ) = \frac{\sqrt{3}}{2} \)
\[
\cos(30^\circ) = \frac{\text{akụkụ dị n'akụkụ}}{\text{hypotenuse}} = \frac{\text{akụkụ dị n'akụkụ}}{8}
\]
\[
\text{Akụkụ Akụkụ} = 8 \cos(30^\circ) = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3} \text{ cm}
\]
Ya mere, ogologo akụkụ nke ọzọ bụ \(4\sqrt{3}\) cm ebe hypotenuse bụ 8 cm.
Ihe atụ nke 3: Iji Trigonometry na nhazi Cartesian
Ajụjụ:
Chọpụta ụkpụrụ nke mmehie, cos, na tan maka isi \( P(3, 4) \) na nhazi Cartesian ma ọ bụrụ na isi ahụ emepụta akụkụ \(\theta\) yana oghere x dị mma na quadrant I.
Azịza:
Nke mbụ, gbakọọ anya nke isi ihe \(P(3, 4)\) site na mmalite (O), nke bụ hypotenuse nke triangle e guzobere.
– Enwere ike ịgbakọ hypotenuse (\(r\)) site na iji usoro Pythagorean:
\[
r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Mgbe ahụ, enwere ike ịgbakọ trigonometry nke isi ihe P dị ka ndị a:
– Sine (mmehie) bụ oke dị n'etiti ordinate (y) na hypotenuse (r):
\[
\sin(\theta) = \frac{y}{r} = \frac{4}{5}
\]
– Cosine (cos) bụ oke dị n'etiti abscissa (x) na hypotenuse (r):
\[
\cos(\theta) = \frac{x}{r} = \frac{3}{5}
\]
– Tangent (tan) bụ oke dị n'etiti ordinate (y) na abscissa (x):
\[
\tan(\theta) = \frac{y}{x} = \frac{4}{3}
\]
Ya mere, ụkpụrụ trigonometric maka isi P bụ:
\[
\sin(\theta) = \frac{4}{5}, \quad \cos(\theta) = \frac{3}{5}, \quad \tan(\theta) = \frac{4}{3}
\]
Ihe atụ Ajụjụ nke 4: Iji njirimara Trigonometric
Ajụjụ:
Ọ bụrụ na \( \sin(\theta) = \frac{3}{5} \), chọta uru nke \( \cos(\theta) \) site na iji njirimara trigonometric \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
Azịza:
Anyị na-amalite site na njirimara trigonometric bụ isi:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
E nyere \( \sin(\theta) = \frac{3}{5} \), wee:
\[
\sin^2(\theta) = \left(\frac{3}{5}\right)^2 = \frac{9}{25}
\]
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
\[
\frac{9}{25} + \cos^2(\theta) = 1
\]
\[
\cos^2(\theta) = 1 – \frac{9}{25} = \frac{25}{25} – \frac{9}{25} = \frac{16}{25}
\]
\[
\cos(\theta) = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}
\]
Uru nke \( \cos(\theta) \) nwere ike ịbụ ihe dị mma ma ọ bụ ihe na-adịghị mma dabere na quadrant nke akụkụ \(\theta\) dị na ya. Mana maka nsogbu a na-enweghị nkọwa quadrant doro anya, anyị na-eji \(\cos(\theta) = \frac{4}{5}\) maka quadrant I ma ọ bụ \( -\frac{4}{5}\) maka quadrants II, III, ma ọ bụ IV.
Mmechi
Ịghọta oke trigonometric dị oke mkpa maka idozi nsogbu dị iche iche gụnyere akụkụ na ogologo na triangles. Site n'ịmụta idozi nsogbu dịka nke dị n'elu, ị nwere ike imeziwanye ikike gị ịgbakọ ma ghọta mmekọrịta dị n'etiti akụkụ na akụkụ na triangles. A na-eji trigonometry eme ihe ọ bụghị naanị na mgbakọ na mwepụ dị ọcha kamakwa na ngalaba sayensị dị iche iche dịka fiziki, injinia, na mbara igwe. Obi dị m ụtọ ime ihe omume!
