Notes de cours - Mathématiques

o
cos sin tan
x∈R(O, −→ i , −→ j)
M OM = 1 (−→ i , −−→ OM) = x[2π]
cos x M
sin x M
tan x=sin x
cos xx6=π/2[π]
cotanx=cos x
sin xx6= 0[π]
tan cotan R
tan x
x∈R
cos2x+ sin2x= 1 .
1 + tan2x=1
cos2x.

cos sin 2π
∀x∈R,cos(x+ 2π) = cos x, sin(x+ 2π) = sin x;
cos sin
∀x∈R,cos(−x) = cos x, sin(−x) = −sin x;
∀x∈R,cos(x+π) = −cos x, sin(x+π) = −sin x
∀x∈R,sin(x+π/2) = cos x, cos(x+π/2) = −sin x
(cos x= 0 ⇔x=π/2 [π]⇔x=π/2−π/2 [2π]
sin x= 0 ⇔x= 0 [π]⇔x= 0 π[2π]
x π/6π/4π/3 1
cos x1√3/2√2/2 1/2 0
sin x0 1/2√2/2√3/2 1
cos(2x)x=π/4 cos(3x)
x=π/6
p0/4p1/4p2/4
p3/4p4/4
tan tan 0 = 0 tan(π/4) = 1

a, b ∈R
cos(a+b) = cos acos b−sin asin b , sin(a+b) = cos asin b+ sin acos b .
ei(a+b)=eiaeib
cos(a+b) = Re((cos a+isin a)(cos b+isin b)) = cos acos b−sin asin b
sin(a+b) = Im((cos a+isin a)(cos b+isin b)) = cos asin b+ cos asin b
a=b
cos(2a) = cos2a−sin2a= 2 cos2a−1 = 1 −2 sin2a ,
sin(2a) = 2 cos asin a .
cos sin
cos(a−b) = cos acos b+ sin asin b, sin(a−b) = sin acos b−cos asin b.
c∈[−1,1] cos x=c
[0, π] arccos c
s∈[−1,1] sin x=s
[−π/2, π/2] arcsin s
t∈Rtan x=t
]−π/2, π/2[ arctan t
R
•cos x=c
x= arccos c[2π]x=−arccos c[2π]
•sin x=s
x= arcsin s[2π]x=π−arcsin s[2π]
•tan x=t
x= arctan t[π]

f(x) = αcos(x) + βsin(x)
α, β ∈Rρ φ
∀x∈R, f(x) = ρcos(x−φ),
ρ φ
α, β ∈Rαcos(x) + βsin(x) = ρcos(x−φ)
ρ=|α+iβ|, φ = Arg(α+iβ) [2π].
ρ=pα2+β2
αcos(x) + βsin(x) = ρα
ρcos x+β ρsin x
cos φ=α/ρ sin φ=β/ρ φ = Arg(α+iβ)
αcos θ+βsin θ ρ cos(θ−φ)
ρ ρ
3 cos x+√3 sin x=−√6
cospθsinqθ
cos(kθ) sin(kθ)
cos θeiθ +e−iθ
2sin θeiθ
−e−iθ
2i
cos(kθ) sin(kθ)
sin θcos2θ=eiθ −e−iθ
2ieiθ +e−iθ 22
···
=1
8iei3θ+eiθ −e−iθ −e−i3θ
=1
8i(2isin(3θ)+2isin(θ))
=sin(3θ) + sin(θ)
4

I=Zπ/2
0
sin2tcos3tdt.
I=2
15
cos(nθ) sin(nθ)
cos(nθ)
cospθsinqθ
(eiθ)n=einθ
(eiθ)n= (cos θ+isin θ)n
cos(nθ) = Re(einθ) sin(nθ) = Im(einθ)
cos(nθ) = Re ((cos θ+isin θ)n)
cos(3θ) = Re (cos θ+isin θ)3
= Re cos3θ+ 3icos2θsin θ−3 cos θsin2θ−isin3θ
= cos3θ−3 cos θsin2θ.
cos cos(3θ)
cos θcos(3θ)
cos(π/18) cos(π/6)