Fa'ata'ita'iga o Fesili e Talanoaina ai Galuega Fa'atino Trigonometric
O galuega faatino a le trigonometric o se vaega taua tele o le matematika, e masani ona aliali mai i vaega eseese o le saienisi, e aofia ai le fisiki, inisinia, ma le saienisi komepiuta. I totonu o lenei tusiga, o le a matou talanoaina ai ni nai faʻataʻitaʻiga o faʻafitauli ma tuʻuina atu se talanoaga loloto o galuega faatino a le trigonometric. O le malamalama i nei faʻataʻitaʻiga, o le a faʻamoemoe o le a faʻamalosia e le au faitau lo latou malamalama ma le mafai ona foia faʻafitauli e aofia ai galuega faatino a le trigonometric.
Fa'atomuaga i Galuega Taulima
O galuega tauave trigonometric e sili ona taatele o le sine (sin), cosine (cos), ma le tangent (tan). O nei galuega tauave e tolu e taua tele i le sootaga i le va o tulimanu ma le umi i tafatolu sa'o, faapea foi i galu ma gatete.
Fua Fa'atatau:
1. Sine (agasala)
\[
\sin(\theta) = \frac{\text{faafeagai}}{\text{hypotenuse}}
\]
2. Kosine (cos)
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
3. Lanu samasama (enaena)
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Fa'asinomaga Trigonometric
– Pitagoras:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
– Faatusatusaga o le tangent ma le sine ma le cosine:
\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\]
– Faasinomaga faaopoopo:
\[
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
\]
\[
\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta)
\]
Se'i o tatou tilotilo i ni fa'ata'ita'iga o fesili ma se talanoaga loloto atu.
Faʻataʻitaʻiga Fesili 1: Fuafuaina o le Taua o Galuega Faʻatino Trigonometric i se Tulimanu Faapitoa
Fesili:
Fuafua ia tau o le sin (30°), cos (45°), ma le tan (60°).
Talanoaga:
E tusa ai ma le laulau o tau faavae o le trigonometric, o loʻo ia i tatou:
– \(\sin(30°) = \frac{1}{2} = 0.5\)
– \(\cos(45°) = \frac{\sqrt{2}}{2} \approx 0.707\)
– \(\tan(60°) = \sqrt{3} \approx 1.732\)
O tau e tolu o loʻo i luga o tau trigonometric ia e masani ona faʻaaogaina, ma e sili ona lelei le taulotoina aua e masani ona aliali mai i fesili.
Fa'ata'ita'iga Fesili 2: Fuafuaina o tulimanu e fa'aaoga ai galuega fa'atino fa'afeagai o le Trigonometric
Fesili:
Afai o le \(\sin(\theta) = 0.5\), fuafua le tau o le \(\theta\).
Talanoaga:
Ina ia maua le tau o le \(\theta\), e manaʻomia ona tatou faʻaaogaina le galuega faʻafeagai o le sine, e pei o le \(\arcsin\) poʻo le \(\sin^{-1}\).
\[
\theta = \sin^{-1}(0.5)
\]
I le va [0°, 360°], o tau talafeagai o \(\theta\) e faapea:
\[
\theta = 30° \text{ ma le } 150°
\]
auā o le \(\sin(30°) = 0.5\) ma le \(\sin(150°) = 0.5\). O lea la, o tau o tulimanu e lua e faʻamalieina o le 30° ma le 150°.
Fa'ata'ita'iga Fesili 3: Fa'aaogaina o Fa'asinomaga Trigonometric
Fesili:
Fa'amaonia fa'asinomaga trigonometric
\[
\sin^2(\theta) + \cos^2(\theta) = 1.
\]
Talanoaga:
O lenei faasinomaga e sau mai le teorama Pythagorean i tafatolu sa'o. Faapea o loo i ai se tafatolu sa'o ma le tulimanu \(\theta\), le itu faafeagai \(a\), le itu e lata ane \(b\), ma le itu e pito i lalo \(c\). Ona,
\[
a^2 + b^2 = c^2.
\]
Afai tatou te vaevaeina itu uma e lua i le \(c^2\), tatou te maua:
\[
\left(\frac{a}{c}\right)^2 + \left(\frac{b}{c}\right)^2 = 1.
\]
Karena
\[
\sin(\theta) = \frac{a}{c} \quad \text{and} \quad \cos(\theta) = \frac{b}{c},
\]
o lea la,
\[
\sin^2(\theta) + \cos^2(\theta) = 1.
\]
O le ala lea tatou te faʻamaonia ai lenei faasinomaga.
Fa'ata'ita'iga Fesili 4: Fa'aaogaina o Galuega Taulima Trigonometric i le Foia o Tafatolu
Fesili:
Ua tu'uina atu le tafatolu ABC ma le tulimanu A 45°, le tulimanu B 60°, ma le itu AB e 10 cm le umi. Saili le umi o itu AC ma le BC.
Talanoaga:
Faaaoga le tulafono sine e sue ai le umi o itu AC ma BC.
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]
Muamua, tatou te maua le tulimanu C:
\[
C = 180° – A – B = 180° – 45° – 60° = 75°.
\]
Fa'atasi ai ma le AB = 10 cm, \(A = 45°\), ma le \(B = 60°\), e mafai ona tatou fa'aogaina le tulafono sine:
\[
\frac{AC}{\sin(60°)} = \frac{10}{\sin(75°)}.
\]
\[
AC = \frac{10 \sin(60°)}{\sin(75°)}.
\]
\[
AC = \frac{10 \times \frac{\sqrt{3}}{2}}{\sin(75°)} = \frac{10 \times \frac{\sqrt{3}}{2}}{\cos(15°)}.
\]
Ua tatou iloa o le \(\cos(15°) = \cos(45° – 30°) = \cos 45° \cos 30° + \sin 45° \sin 30° = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2}\).
\[
\cos(15°) = \frac{\sqrt{6} + \sqrt{2}}{4}.
\]
O lena la:
\[
AC = \frac{10 \times \frac{\sqrt{3}}{2}}{\frac{\sqrt{6} + \sqrt{2}}{4}} = \frac{10 \times 2\sqrt{3}}{\sqrt{6} + \sqrt{2}} \approx 10.39 \text{ cm}.
\]
I se auala talitutusa, e mafai ona tatou maua le BC:
\[
\frac{BC}{\sin(45°)} = \frac{10}{\sin(75°)}.
\]
\[
BC = \frac{10 \sin(45°)}{\sin(75°)} \approx 8.66 \text{ cm}.
\]
I le faaiuga o lenei tusiga, ua matou talanoaina ni nai faataitaiga o faafitauli ma a latou talanoaga e faatatau i galuega faatino trigonometric. Faatasi ai ma le faataitai faifai pea ma le malamalama lelei i fua faavae, faasinomaga trigonometric, ma a latou faaaogaina i tafatolu, e faamoemoeina le au faitau e sili atu ona malamalama i lenei mataupu. O galuega faatino trigonometric o ni meafaigaluega taua, e le gata i le matematika ae faapea foi i mataupu eseese e faalagolago i le auiliiliga o tulimanu ma umi. Matou te faamoemoe o lenei tusiga o se faasino aoga mo le au faitau.
