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Formulaire de trigonométrie circulaire A 1 B x M H K cos(x) sin(x) tan(x) cotan( x) cos(x) = abscisse de M sin(x) = ordonnée de M tan(x) = AH cotan(x) = BK

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La fonction x ↦→ cos(x) est définie sur R, 2π-périodique et paire Formules d' addition cos(a + b) = cos(a) cos(b) − sin(a) sin(b) cos(a − b) = cos(a) cos(b) + 

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Formules d'addition cos(a + b) = cos(a) cos(b) − sin(a) sin(b) cos(a − b) = cos(a) cos(b) + sin(a) sin(b) sin(a + b) = sin(a) cos(b) + cos(a) sin(b) sin(a − b) 

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19 nov 2014 · 2 Formules d'addition et de différence Rappelons les formules d'addition : cos(a + b) = cosacosb - sinasinb sin(a + b) = sinacosb + sinbcosa

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On définit les fonctions cos, sin et tan par les formules cos(x) = eix + e−ix 2 = Re(eix), sin(x) = (cos(a) + i cos(b))n = cos(na) + i sin(na) On peut linéariser les  

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1 + cot2 θ = cosec2θ (2) tan2 θ + 1 = sec2 θ (3) Note that (2) = (1)/ sin2 θ and (3 ) = (1)/ cos2 θ Compound-angle formulae cos(A + B) = cos A cos B − sin A sin B

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cos(π 2 − a) = sin a sin (π 2 − a) = cosa tan(π 2 − a) = 1 tana 4 Formules d' addition cos (a + b) = cosa cosb − sin a sin b cos (a − b) = cosa cosb + sin a sin  

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Or ei(a+b)=cos(a)cos(b)sin(a)sin(b)+i(sin(b)cos(a)+sin(a)cos(b)) Donc sin (a+b)= sin (a)cos(b)+sin (b)cos(a) c) cos(ab) et sin(ab) : Ici il suffit de remplacer b par b

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cos(x) = eix + e-ix 2 et sin(x) = eix − e-ix 2i cos2(a) + sin2(a)=1 On est sur le cercle de rayon 1, d'équation x2 + y2 = 1 cos(a + b) = cos(a) cos(b) − sin(a) sin( b)

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On définit les fonctions cosinus, sinus et tangente, notées cos, sin et tan telles que cos(a + b) = cosa cosb − sin a sin b sin(a + b) = sin a cosb + cosa sinb

pdf Trigonometric Identities - The University of Liverpool

Pythagoras’s theorem sin2 + cos2 = 1 (1) 1 + cot2 = cosec2 (2) tan2 + 1 = sec2 (3) Note that (2) = (1)=sin2 and (3) = (1)=cos Compound-angle formulae cos(A+ B) = cosAcosB sinAsinB (4) cos(A B) = cosAcosB+ sinAsinB (5) sin(A+ B) = sinAcosB+ cosAsinB (6) sin(A B) = sinAcosB cosAsinB (7) tan(A+ B) = tanA+ tanB 1 tanAtanB (8) tan(A B) = tanA

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Trigonometric Identities

Pythagoras's theorem

sin

2+ cos2= 1 (1)

1 + cot

2= cosec2(2)

tan

2+ 1 = sec2(3)

Note that (2) = (1)=sin2and (3) = (1)=cos2.

Compound-angle formulae

cos(A+B) = cosAcosBsinAsinB(4) cos(AB) = cosAcosB+ sinAsinB(5) sin(A+B) = sinAcosB+ cosAsinB(6) sin(AB) = sinAcosBcosAsinB(7) tan(A+B) =tanA+ tanB1tanAtanB(8) tan(AB) =tanAtanB1 + tanAtanB(9) cos2= cos2sin2= 2cos21 = 12sin2(10) sin2= 2sincos(11) tan2=2tan1tan2(12) Note that you can get (5) from (4) by replacingBwithB, and using the fact that cos(B) = cosB(cos is even) and sin(B) =sinB(sin is odd). Similarly (7) comes from (6). (8) is obtained by dividing (6) by (4) and dividing top and bottom by cosAcosB, while (9) is obtained by dividing (7) by (5) and dividing top and bottom by cosAcosB. (10), (11), and (12) are special cases of (4), (6), and (8) obtained by putting A=B=.

Sum and product formulae

cosA+ cosB= 2cosA+B2cosAB2(13) cosAcosB=2sinA+B2sinAB2(14) sinA+ sinB= 2sinA+B2cosAB2(15) sinAsinB= 2cosA+B2sinAB2(16) Note that (13) and (14) come from (4) and (5) (to get (13), use (4) to expand cosA= cos( A+B2+AB2) and (5) to expand cosB= cos(A+B2AB2), and add the results).

Similarly (15) and (16) come from (6) and (7).

Thusyou only need to remember (1), (4), and (6): the other identities can be derived from these.quotesdbs_dbs18.pdfusesText_24