5.2 Reduksieformule

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Reduksie formule

Enige trigonometriese funksie wat se argument $90^{\circ} \pm θ$ , $180^{\circ} \pm θ$ , $270^{\circ} \pm θ$ is en $360^{\circ} \pm θ$ (dus $- θ$ ) kan eenvoudig geskryf word in terme van $θ$ . Byvoorbeeld, jy kon opgemerk het dat die cosinus-grafiek identies is aan die sinus-grafiek behalwe vir 'n fase verskuiwing van $90^{\circ}$ . Dit kan dus afgelei word dat $sin (90^{\circ} + θ) = cos θ$ .

Funksie waardes van $180^{\circ} \pm θ$

Ondersoek : reduksie formules vir funksie waardes van $180^{\circ} \pm θ$

Funksie Waardes van $(180^{\circ} - θ)$
1. In die figuur lê P en P' op die sirkel met radius 2. OP vorm 'n hoek $θ = 30^{\circ}$ met die $x$ -as. P het dus die koördinate $(\sqrt{3}; 1)$ . As P' die refleksie van P om die $y$ -as (of die lyn $x = 0$ ) is, gebruik simmetrie om die koördinate van P' neer te skryf.
2. Gee waardes vir $sin θ$ , $cos θ$ en $tan θ$ .
3. Deur gebruik te maak van die koördinate van P' bepaal: $sin (180^{\circ} - θ)$ , $cos (180^{\circ} - θ)$ en $tan (180^{\circ} - θ)$ .
1. (d) Probeer om uit die resultate 'n verhouding tussen die funksie waardes van $(180^{\circ} - θ)$ en $θ$ te bepaal.
Funksie waardes van $(180^{\circ} + θ)$
1. In die figuur lê P en P' op die sirkel met radius 2. OP vorm 'n hoek $θ = 30^{\circ}$ met die $x$ -as. P het dus die koördinate $(\sqrt{3}; 1)$ . As P' die refleksie van P om die $y$ -as (of die lyn $x = 0$ ) is, gebruik simmetrie om die koördinate van P' neer te skryf.
2. Gee waardes vir $sin θ$ , $cos θ$ en $tan θ$ .
3. Deur gebruik te maak van die koördinate van P' bepaal: $sin (180^{\circ} + θ)$ , $cos (180^{\circ} + θ)$ en $tan (180^{\circ} + θ)$ .

Ondersoek: reduksie formules vir funksie waardes van $360^{\circ} \pm θ$

Funksie waardes van $(360^{\circ} - θ)$
1. In die figuur lê P en P' op die sirkel met radius 2. OP vorm 'n hoek $θ = 30^{\circ}$ met die $x$ -as. P het dus die koördinate $(\sqrt{3}; 1)$ . As P' die refleksie van P om die $y$ -as (of die lyn $x = 0$ ) is, gebruik simmetrie om die koördinate van P' neer te skryf.
2. Gee waardes vir $sin θ$ , $cos θ$ en $tan θ$ .
3. Deur gebruik te maak van die koördinate van P' bepaal: $sin (360^{\circ} - θ)$ , $cos (360^{\circ} - θ)$ en $tan (360^{\circ} - θ)$ .

Dit is moontlik om 'n hoek groter as $360^{\circ}$ te hê. Die hoek voltooi een omwenteling van $360^{\circ}$ en dan gaan dit voort om die vereiste hoek te gee. Ons kry die volgende resultate:

\begin{matrix} sin (360^{\circ} + θ) & = & sin θ \\ cos (360^{\circ} + θ) & = & cos θ \\ tan (360^{\circ} + θ) & = & tan θ \end{matrix}

Neem ook kennis, dat as $k$ enige heelgetal is, dan is

\begin{matrix} sin (k 360^{\circ} + θ) & = & sin θ \\ cos (k 360^{\circ} + θ) & = & cos θ \\ tan (k 360^{\circ} + θ) & = & tan θ \end{matrix}

Skryf $sin 293^{\circ}$ as die funksie van 'n skerphoek.

Ons let op dat $293^{\circ} = 360^{\circ} - 67^{\circ}$ dus
$\begin{matrix} sin 293^{\circ} & = & sin (360^{\circ} - 67^{\circ}) \\ = & - sin 67^{\circ} \end{matrix}$

waar ons die feit gebruik het dat $sin (360^{\circ} - θ) = - sin θ$ . Toets, met behulp van jou sakrekenaar, dat hierdie waardes wel reg is:

$\begin{matrix} sin 293^{\circ} & = & - 0, 92 \dots \\ - sin 67^{\circ} & = & - 0, 92 \dots \end{matrix}$

Evalueer sonder 'n sakrekenaar:

{tan}^{2} 210^{\circ} - (1 + cos 120^{\circ}) {sin}^{2} 225^{\circ}

Herskryf elke hoek as 'n skerphoek
$\begin{matrix} {tan}^{2} 210^{\circ} - (1 + cos 120^{\circ}) {sin}^{2} 225^{\circ} \\ = & {[tan (180^{\circ} + 30^{\circ})]}^{2} - [1 + cos (180^{\circ} - 60^{\circ})] \cdot {[sin (180^{\circ} + 45^{\circ})]}^{2} \\ = & {(tan 30^{\circ})}^{2} - [1 + (- cos 60^{\circ})] \cdot {(- sin 45^{\circ})}^{2} \\ = & {(\frac{1}{\sqrt{3}})}^{2} - (1 - \frac{1}{2}) \cdot {(- \frac{1}{\sqrt{2}})}^{2} \\ = & \frac{1}{3} - (\frac{1}{2}) (\frac{1}{2}) \\ = & \frac{1}{3} - \frac{1}{4} = \frac{1}{12} \end{matrix}$

Reduksie formules

Skryf hierdie vergelykings as 'n funksie van slegs θ :
1. $sin (180^{\circ} - θ)$
2. $cos (180^{\circ} - θ)$
3. $cos (360^{\circ} - θ)$
4. $cos (360^{\circ} + θ)$
5. $tan (180^{\circ} - θ)$
6. $cos (360^{\circ} + θ)$
Skryf die volgende trig funksies as 'n funksie van' n skerphoek:
1. $sin 163^{\circ}$
2. $cos 327^{\circ}$
3. $tan 248^{\circ}$
4. $cos 213^{\circ}$
Bepaal die volgende sonder die gebruik van 'n sakrekenaar:
1. $tan 150^{\circ} . sin 30^{\circ} + cos 330^{\circ}$
2. $tan 300^{\circ} . cos 120^{\circ}$
3. $(1 - cos 30^{\circ}) (1 - sin 210^{\circ})$
4. $cos 780^{\circ} + sin 315^{\circ} . tan 420^{\circ}$
Bepaal die volgende deur dit te herlei na 'n skerphoek en met behulp van spesiale hoeke. Moenie 'n sakrekenaar gebruik nie:
1. $cos 300^{\circ}$
2. $sin 135^{\circ}$
3. $cos 150^{\circ}$
4. $tan 330^{\circ}$
5. $sin 120^{\circ}$
6. ${tan}^{2} 225^{\circ}$
7. $cos 315^{\circ}$
8. ${sin}^{2} 420^{\circ}$
9. $tan 405^{\circ}$
10. $cos 1020^{\circ}$
11. ${tan}^{2} 135^{\circ}$
12. $1 - {sin}^{2} 210^{\circ}$

Funksie waardes van $(- θ)$

Wanneer die argument van 'n trigonometriese funksie $(- θ)$ is kan $360^{\circ}$ by voeg sonder om die resultaat te verander. So vir sinus en kosinus het ons

sin (- θ) = sin (360^{\circ} - θ) = - sin θ

cos (- θ) = cos (360^{\circ} - θ) = cos θ

Funksie waardes van $90^{\circ} \pm θ$

Ondersoek: reduksie formules vir funksie waardes van $90^{\circ} \pm θ$

Funksie waardes van $(90^{\circ} - θ)$
1. In die figuur lê P en P' op die sirkel met radius 2. OP vorm 'n hoek $θ = 30^{\circ}$ met die $x$ -as. P het dus die koördinate $(\sqrt{3}; 1)$ . As P' die refleksie van P om die $y$ -as (of die lyn $x = 0$ ) is, gebruik simmetrie om die koördinate van P' neer te skryf.
2. Gee waardes vir $sin θ$ , $cos θ$ en $tan θ$ .
3. Deur gebruik te maak van die koördinate van P' bepaal: $sin (90^{\circ} - θ)$ , $cos (90^{\circ} - θ)$ en $tan (90^{\circ} - θ)$ .
Funksie waardes van $(90^{\circ} + θ)$
1. In die figuur lê P en P' op die sirkel met radius 2. OP vorm 'n hoek $θ = 30^{\circ}$ met die $x$ -as. P het dus die koördinate $(\sqrt{3}; 1)$ . As P' die refleksie van P om die $y$ -as (of die lyn $x = 0$ ) is, gebruik simmetrie om die koördinate van P' neer te skryf.
2. Gee waardes vir $sin θ$ , $cos θ$ en $tan θ$ .
3. Deur gebruik te maak van die koördinate van P' bepaal: $sin (90^{\circ} + θ)$ , $cos (90^{\circ} + θ)$ en $tan (90^{\circ} + θ)$ .

Komplementêre hoeke is positiewe skerphoeke wat gelyk is aan $90^{\circ}$ . Bv. $20^{\circ}$ en $70^{\circ}$ is komplimentere hoeke.

Sinus en kosinus staan bekend as ko-funksies . Twee funksies word ko-funksies genoem indien $f (A) = g (B)$ whenever $A + B = 90^{\circ}$ (m.a.w. $A$ en $B$ is komplimentere hoeke). Die ander trig ko-funksies is secans en cosecans, en die tangens en cotangens.

Die funksie waarde van 'n hoek is gelyk aan die ko-funksie van sy komplement (die ko-ko-reël).

Dus het ons vir sinus en kosinus as

\begin{matrix} sin (90^{\circ} - θ) & = & cos θ \\ cos (90^{\circ} - θ) & = & sin θ \end{matrix}

Skryf elk van die volgende in terme van $40^{\circ}$ deur gebruik te maak van $sin (90^{\circ} - θ) = cos θ$ en $cos (90^{\circ} - θ) = sin θ$ .

$cos 50^{\circ}$
$sin 320^{\circ}$
$cos 230^{\circ}$

Ons gebruik wat ons sover geleer het:
1. $cos 50^{\circ} = cos (90^{\circ} - 40^{\circ}) = sin 40^{\circ}$
2. $sin 320^{\circ} = sin (360^{\circ} - 40^{\circ}) = - sin 40^{\circ}$
3. $cos 230^{\circ} = cos (180^{\circ} + 50^{\circ}) = - cos 50^{\circ} = - cos (90^{\circ} - 40^{\circ}) = - sin 40^{\circ}$

Funksie waardes van $(θ - 90^{\circ})$

$sin (θ - 90^{\circ}) = - cos θ$ and $cos (θ - 90^{\circ}) = sin θ$ .

Hierdie resultate kan as volg bewys word:

\begin{matrix} sin (θ - 90^{\circ}) & = & sin [- (90^{\circ} - θ)] \\ = & - sin (90^{\circ} - θ) \\ = & - cos θ \end{matrix}

en so ook vir $cos (θ - 90^{\circ}) = sin θ$

Opsomming

Die volgende opsomming kan gemaak word

tweede kwadrant $(180^{\circ} - θ)$ or $(90^{\circ} + θ)$	eerste kwadrant $(θ)$ or $(90^{\circ} - θ)$
$sin (180^{\circ} - θ) = + sin θ$	alle trig funksies is positief
$cos (180^{\circ} - θ) = - cos θ$	$sin (360^{\circ} + θ) = sin θ$
$tan (180^{\circ} - θ) = - tan θ$	$cos (360^{\circ} + θ) = cos θ$
$sin (90^{\circ} + θ) = + cos θ$	$tan (360^{\circ} + θ) = tan θ$
$cos (90^{\circ} + θ) = - sin θ$	$sin (90^{\circ} - θ) = sin θ$
	$cos (90^{\circ} - θ) = cos θ$
derde kwadrant $(180^{\circ} + θ)$	vierde kwadrant $(360^{\circ} - θ)$
$sin (180^{\circ} + θ) = - sin θ$	$sin (360^{\circ} - θ) = - sin θ$
$cos (180^{\circ} + θ) = - cos θ$	$cos (360^{\circ} - θ) = + cos θ$
$tan (180^{\circ} + θ) = + tan θ$	$tan (360^{\circ} - θ) = - tan θ$

Hierdie reduksie formules geld vir enige hoek $θ$ . Vir gerief, werk ons gewoonlik met $θ$ asof dit 'n skerphoek is, m.a.w. $0^{\circ} < θ < 90^{\circ}$ .
By die bepaling van die funksie waardes van $180^{\circ} \pm θ$ , $360^{\circ} \pm θ$ and $- θ$ verander die funksies nooit.
By die bepaling van die funksie waardes van $90^{\circ} \pm θ$ and $θ - 90^{\circ}$ verander die funksies na sy ko-funksies (ko-ko-reel).

Funksie waardes van $(270^{\circ} \pm θ)$

Hoeke in die derde en vierde kwadrante kan geskryf word as $270^{\circ} \pm θ$ met $θ$ 'n skerphoek. Soortgelyke reëls as bogenoemde is van toepassing. Ons kry

derde kwadrant $(270^{\circ} - θ)$	vierde kwadrant $(270^{\circ} + θ)$
$sin (270^{\circ} - θ) = - cos θ$	$sin (270^{\circ} + θ) = - cos θ$
$cos (270^{\circ} - θ) = - sin θ$	$cos (270^{\circ} + θ) = + sin θ$

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Source: OpenStax, Siyavula textbooks: wiskunde (graad 11). OpenStax CNX. Sep 20, 2011 Download for free at http://cnx.org/content/col11339/1.4

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5.2 Reduksieformule

Reduksie formule

Funksie waardes van 180 ∘ ± θ

Ondersoek : reduksie formules vir funksie waardes van 180 ∘ ± θ

Ondersoek: reduksie formules vir funksie waardes van 360 ∘ ± θ