Fa'ata'ita'iga o Fesili ma Talanoaga o Mea e Maua Mai i Galuega Tau Trigonometric
O le derivative o se manatu faavae i le calculus, e masani ona faʻaaogaina e faʻamatalaina ai le fua faatatau o le suiga o se galuega faatino. I le tulaga o galuega faatino trigonometric, e fesoasoani le derivative ia i tatou e malamalama pe faʻapefea ona aʻafia e suiga i tulimanu le taua o le galuega faatino. I totonu o lenei tusiga, o le a tatou talanoaina ai ni faʻataʻitaʻiga o faʻafitauli ma fofo e fesoʻotaʻi ma derivatives o galuega faatino trigonometric.
Fa'atomuaga i Galuega Taulima
O galuega faatino autu o le trigonometric e masani ona faʻaaogaina e aofia ai le sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (cosec), ma le cotangent (cot). O galuega faatino taitasi e iai se mea e maua mai ai:
1. \( \frac{d}{dx} \sin(x) = \cos(x) \)
2. \( \frac{d}{dx} \cos(x) = -\sin(x) \)
3. \( \frac{d}{dx} \tan(x) = \sec^2(x) \)
4. \( \frac{d}{dx} \sec(x) = \sec(x) \tan(x) \)
5. \( \frac{d}{dx} \csc(x) = -\csc(x) \cot(x) \)
6. \( \frac{d}{dx} \cot(x) = -\csc^2(x) \)
Faatasi ai ma lenei malamalamaaga faavae, e mafai ona tatou agai atu i ni faʻataʻitaʻiga loloto o faʻafitauli ma fofo.
Fa'ata'ita'iga Fesili 1: Fa'asologa o le Sine Function
Soal
Saili le derivative o le function \( f(x) = 3\sin(x) \).
Penyelesaian
Mo le sailia o le derivative o le function \( f(x) = 3\sin(x) \), e mafai ona tatou faʻaaogaina tulafono faavae o derivatives faʻapea foʻi ma constants i le calculus. O le derivative o le \( \sin(x) \) o le \( \cos(x) \).
\[
f'(x) = 3 \cdot \frac{d}{dx} \sin(x) = 3\cos(x)
\]
O lea la, o le fua fa'atatau o le \( f(x) = 3\sin(x) \) o le \( 3\cos(x) \).
Faʻataʻitaʻiga 2: Tuufaatasiga o Galuega Sine ma Cosine
Soal
Saili le fua fa'atatau o le galuega faatino \( g(x) = 2\sin(x) + 4\cos(x) \).
Penyelesaian
Ina ia maua le derivative o le function \( g(x) = 2\sin(x) + 4\cos(x) \), e mafai ona tatou faʻaaogaina tulafono faʻavae o derivative ma faʻailoa mai derivative taʻitasi o \( \sin(x) \) ma le \( \cos(x) \).
\[
g'(x) = 2 \cdot \frac{d}{dx} \sin(x) + 4 \cdot \frac{d}{dx} \cos(x)
\]
Ua matou iloa:
\[
\frac{d}{dx} \sin(x) = \cos(x)
\]
\[
\frac{d}{dx} \cos(x) = -\sin(x)
\]
O lena la:
\[
g'(x) = 2 \cos(x) + 4(-\sin(x)) = 2\cos(x) – 4\sin(x)
\]
O lea la, o le fua fa'atatau o le \( g(x) = 2\sin(x) + 4\cos(x) \) e \( 2\cos(x) – 4\sin(x) \).
Faʻataʻitaʻiga 3: Galuega Faʻatafafā o le Sine
Soal
Saili le fua fa'atatau o le galuega faatino \( h(x) = (\sin(x))^2 \).
Penyelesaian
Mo le su'eina o le derivative o le function \( h(x) = (\sin(x))^2 \), e mafai ona tatou fa'aaogaina le chain rule.
Muamua, tatou setiina le \( u = \sin(x) \), ina ia \( h(x) = u^2 \).
Ua tatou iloa o le derivative o le \( u^2 \) e faatatau i le \( u \) o le \( 2u \), ma o le derivative o le \( u \) e faatatau i le \( x \) o le \( \cos(x) \).
O lea la,
\[
\frac{d}{dx} (\sin(x))^2 = 2 (\sin(x)) \cdot \cos(x)
\]
O lea la, o le fua fa'atatau o le \( h(x) = (\sin(x))^2 \) o le \( 2\sin(x)\cos(x) \).
Fa'ata'ita'iga Fesili 4: Galuega Fa'atino o le Tangent
Soal
Saili le fua fa'atatau o le galuega faatino \( f(x) = \tan(x) \).
Penyelesaian
Mo le su'eina o le derivative o le \( f(x) = \tan(x) \), matou te fa'aaogaina le fa'amatalaga o le derivative o le tangent.
\[
\frac{d}{dx} \tan(x) = \sec^2(x)
\]
O lea la, o le derivative o le \( f(x) = \tan(x) \) o le \( \sec^2(x) \).
Faataitaiga 5: Tuufaatasiga o Galuega Faatino a le Tangent ma le Secant
Soal
Saili le derivative o le function \( p(x) = \tan(x)\sec(x) \).
Penyelesaian
Ina ia maua le fua fa'atatau o le oloa o galuega faatino e lua, e tatau ona tatou fa'aaogaina le tulafono o le oloa.
\[
(fg)' = f'g + fg'
\]
O fea \( f(x) = \tan(x) \) ma \( g(x) = \sec(x) \).
Ua matou iloa:
\[
f'(x) = \sec^2(x)
\]
\[
g'(x) = \sec(x)\tan(x)
\]
O lena la:
\[
p'(x) = \tan(x) \cdot \sec(x) \tan(x) + \sec(x) \cdot \sec^2(x)
\]
\[
p'(x) = \sec^2(x) \tan^2(x) + \sec^3(x)
\]
O lea la, o le fa'aliliuga o le \( p(x) = \tan(x)\sec(x) \) e \( \sec^2(x) \tan^2(x) + \sec^3(x) \).
Fa'ata'ita'iga Fesili 6: Galuega Fa'atino Cosecant ma Cotangent
Soal
Saili le fua fa'atatau o le galuega faatino \( q(x) = \csc(x) – \cot(x) \).
Penyelesaian
Mo le su'eina o le derivative o le \( q(x) = \csc(x) – \cot(x) \), matou te fa'aaogaina fa'amatalaga o le derivative o le cosecant ma le cotangent.
\[
\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)
\]
\[
\frac{d}{dx} \cot(x) = -\csc^2(x)
\]
O lena la:
\[
q'(x) = -\csc(x)\cot(x) – (-\csc^2(x))
\]
\[
q'(x) = -\csc(x)\cot(x) + \csc^2(x)
\]
O lea la, o le fua fa'atatau o le \( q(x) = \csc(x) – \cot(x) \) e \( -\csc(x)\cot(x) + \csc^2(x) \).
I'uga
I totonu o lenei tusiga, ua matou talanoaina ai faʻataʻitaʻiga ma fofo eseese e fesoʻotaʻi ma mea e maua mai i galuega faatino trigonometric. Mai galuega faatino faavae e pei o le sine ma le cosine, i tuʻufaʻatasiga sili atu ona lavelave e pei o le oloa o le tangent ma le secant, ma mea e maua mai i le cosecant ma le cotangent. O le malamalama i mea e maua mai i galuega faatino trigonometric e le gata ina aoga i le matematika mama ae e lautele foi lona faʻaaogaina i le fisiki, inisinia, ma isi vaega eseese e faʻaaogaina ai suiga o galuega faatino ma fua faatatau o suiga.
O le fa'ata'ita'iina atili o fa'afitauli, o le a fa'aleleia atili ai lo tatou malamalama i mea e maua mai i galuega fa'atino trigonometric. E fa'amoemoe o lenei tusiga e fesoasoani ia te oe e malamalama ai i le manatu ma fa'aoga o mea e maua mai i galuega fa'atino trigonometric!
