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Reduksie formule

Enige trigonometriese funksie wat se argument 90 ± θ , 180 ± θ , 270 ± θ is en 360 ± 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 . Dit kan dus afgelei word dat sin ( 90 + θ ) = cos θ .

Funksie waardes van 180 ± θ

Ondersoek : reduksie formules vir funksie waardes van 180 ± θ

  1. Funksie Waardes van ( 180 - θ )
    1. In die figuur lê P en P' op die sirkel met radius 2. OP vorm 'n hoek θ = 30 met die x -as. P het dus die koördinate ( 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 - θ ) , cos ( 180 - θ ) en tan ( 180 - θ ) .
    1. Probeer om uit die resultate 'n verhouding tussen die funksie waardes van ( 180 - θ ) en θ te bepaal.
  2. Funksie waardes van ( 180 + θ )
    1. In die figuur lê P en P' op die sirkel met radius 2. OP vorm 'n hoek θ = 30 met die x -as. P het dus die koördinate ( 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 + θ ) , cos ( 180 + θ ) en tan ( 180 + θ ) .

Ondersoek: reduksie formules vir funksie waardes van 360 ± θ

  1. Funksie waardes van ( 360 - θ )
    1. In die figuur lê P en P' op die sirkel met radius 2. OP vorm 'n hoek θ = 30 met die x -as. P het dus die koördinate ( 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 - θ ) , cos ( 360 - θ ) en tan ( 360 - θ ) .

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

sin ( 360 + θ ) = sin θ cos ( 360 + θ ) = cos θ tan ( 360 + θ ) = tan θ

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

sin ( k 360 + θ ) = sin θ cos ( k 360 + θ ) = cos θ tan ( k 360 + θ ) = tan θ

Skryf sin 293 as die funksie van 'n skerphoek.

  1. sin 293 = sin ( 360 - 67 ) = - sin 67

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

    sin 293 = - 0 , 92 - sin 67 = - 0 , 92

Evalueer sonder 'n sakrekenaar:

tan 2 210 - ( 1 + cos 120 ) sin 2 225
  1. tan 2 210 - ( 1 + cos 120 ) sin 2 225 = [ tan ( 180 + 30 ) ] 2 - [ 1 + cos ( 180 - 60 ) ] · [ sin ( 180 + 45 ) ] 2 = ( tan 30 ) 2 - [ 1 + ( - cos 60 ) ] · ( - sin 45 ) 2 = 1 3 2 - 1 - 1 2 · - 1 2 2 = 1 3 - 1 2 1 2 = 1 3 - 1 4 = 1 12

Reduksie formules

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

Funksie waardes van ( - θ )

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

sin ( - θ ) = sin ( 360 - θ ) = - sin θ
cos ( - θ ) = cos ( 360 - θ ) = cos θ

Funksie waardes van 90 ± θ

Ondersoek: reduksie formules vir funksie waardes van 90 ± θ

  1. Funksie waardes van ( 90 - θ )
    1. In die figuur lê P en P' op die sirkel met radius 2. OP vorm 'n hoek θ = 30 met die x -as. P het dus die koördinate ( 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 - θ ) , cos ( 90 - θ ) en tan ( 90 - θ ) .
  2. Funksie waardes van ( 90 + θ )
    1. In die figuur lê P en P' op die sirkel met radius 2. OP vorm 'n hoek θ = 30 met die x -as. P het dus die koördinate ( 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 + θ ) , cos ( 90 + θ ) en tan ( 90 + θ ) .

Komplementêre hoeke is positiewe skerphoeke wat gelyk is aan 90 . Bv. 20 en 70 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 (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

sin ( 90 - θ ) = cos θ cos ( 90 - θ ) = sin θ

Skryf elk van die volgende in terme van 40 deur gebruik te maak van sin ( 90 - θ ) = cos θ en cos ( 90 - θ ) = sin θ .

  1. cos 50
  2. sin 320
  3. cos 230
    1. cos 50 = cos ( 90 - 40 ) = sin 40
    2. sin 320 = sin ( 360 - 40 ) = - sin 40
    3. cos 230 = cos ( 180 + 50 ) = - cos 50 = - cos ( 90 - 40 ) = - sin 40

Funksie waardes van ( θ - 90 )

sin ( θ - 90 ) = - cos θ and cos ( θ - 90 ) = sin θ .

Hierdie resultate kan as volg bewys word:

sin ( θ - 90 ) = sin [ - ( 90 - θ ) ] = - sin ( 90 - θ ) = - cos θ

en so ook vir cos ( θ - 90 ) = sin θ

Opsomming

Die volgende opsomming kan gemaak word

tweede kwadrant ( 180 - θ ) or ( 90 + θ ) eerste kwadrant ( θ ) or ( 90 - θ )
sin ( 180 - θ ) = + sin θ alle trig funksies is positief
cos ( 180 - θ ) = - cos θ sin ( 360 + θ ) = sin θ
tan ( 180 - θ ) = - tan θ cos ( 360 + θ ) = cos θ
sin ( 90 + θ ) = + cos θ tan ( 360 + θ ) = tan θ
cos ( 90 + θ ) = - sin θ sin ( 90 - θ ) = sin θ
cos ( 90 - θ ) = cos θ
derde kwadrant ( 180 + θ ) vierde kwadrant ( 360 - θ )
sin ( 180 + θ ) = - sin θ sin ( 360 - θ ) = - sin θ
cos ( 180 + θ ) = - cos θ cos ( 360 - θ ) = + cos θ
tan ( 180 + θ ) = + tan θ tan ( 360 - θ ) = - tan θ
  1. Hierdie reduksie formules geld vir enige hoek θ . Vir gerief, werk ons gewoonlik met θ asof dit 'n skerphoek is, m.a.w. 0 < θ < 90 .
  2. By die bepaling van die funksie waardes van 180 ± θ , 360 ± θ and - θ verander die funksies nooit.
  3. By die bepaling van die funksie waardes van 90 ± θ and θ - 90 verander die funksies na sy ko-funksies (ko-ko-reel).

Funksie waardes van ( 270 ± θ )

Hoeke in die derde en vierde kwadrante kan geskryf word as 270 ± θ met θ 'n skerphoek. Soortgelyke reëls as bogenoemde is van toepassing. Ons kry

derde kwadrant ( 270 - θ ) vierde kwadrant ( 270 + θ )
sin ( 270 - θ ) = - cos θ sin ( 270 + θ ) = - cos θ
cos ( 270 - θ ) = - sin θ cos ( 270 + 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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