Trigonométrie - Classe B/L

xcos(x) sin(x)
∀x∈R,cos(x) = eix =eix +e−ix
2∀x∈R,sin(x) = eix =eix −e−ix 2i
cos sin R
∀x∈R,−16cos(x)61−16sin(x)61 cos2(x) + sin2(x)=1
[−1,1]
x, y ∈R
cos(x) = cos(y)⇐⇒ ∃k∈Z/ x =y+ 2kπ
∃k∈Z/ x =−y+ 2kπ
sin(x) = sin(y)⇐⇒ ∃k∈Z/ x =y+ 2kπ
∃k∈Z/ x = (π−y)+2kπ

cos(x)=0 ⇐⇒ ∃k∈Z/ x =π
2+kπ sin(x)=0 ⇐⇒ ∃k∈Z/ x =kπ
cos(x)=1 ⇐⇒ ∃k∈Z/ x = 2kπ sin(x)=1 ⇐⇒ ∃k∈Z/ x =π
2+ 2kπ
cos(x) = −1⇐⇒ ∃k∈Z/ x =π+ 2kπ sin(x) = −1⇐⇒ ∃k∈Z/ x =−π
2+ 2kπ
tan tan(x) = sin(x)
cos(x)
cos(x)6= 0 R\π
2+kπ, k ∈Z
x0π
6
π
4
π
3
π
2
cos(x) 1 √3
2
√2
2
1
20
sin(x) 0 1
2
√2
2
√3
21
tan(x) 0 √3
31√3??
2πR
2πR
π
∀x∈R,cos(−x) = cos(x) cos(x+ 2π) = cos(x)
∀x∈R,sin(−x) = −sin(x) sin(x+ 2π) = sin(x)
∀x∈R\π
2+kπ, k ∈Z,tan(−x) = −tan(x) tan(x+π) = tan(x)
x
cos(x+π) = −cos(x) sin(x+π) = −sin(x) tan(x+π) = tan(x)
cos(π−x) = −cos(x) sin(π−x) = sin(x) tan(π−x) = −tan(x)
cos π
2−x= sin(x) sin π 2−x= cos(x) tan π 2−x=1 tan(x)
cos x+π
2=−sin(x) sin x+π 2= cos(x) tan x+π 2=−1 tan(x)

a b
cos(a+b) = cos(a) cos(b)−sin(a) sin(b) sin(a+b) = sin(a) cos(b) + cos(a) sin(b)
cos(a+b) = (ei(a+b)) sin(a+b) = (ei(a+b))
ei(a+b)=eiaeib = (cos(a) + isin(a))(cos(b) + isin(b))
= (cos(a) cos(b)−sin(a) sin(b)) + i(sin(a) cos(b) + cos(a) sin(b))
ei(a+b)
cos(a+b) = cos(a) cos(b)−sin(a) sin(b) sin(a+b) = sin(a) cos(b)+cos(a) sin(b)
a b
cos(a) + cos(b) = 2 cos a−b
2cos a+b 2,sin(a) + sin(b) = 2 cos a−b 2sin a+b 2
cos(a) + cos(b) = (eia) + (eib) = (eia +eib) sin(a) + sin(b) = (eia +eib)
eia +eib =eia+b
2eia−b 2+eib−a 2=eia+b 22 cos a−b
2
= 2 cos a−b
2cos a+b 2+i2 cos a−b 2sin a+b 2
eia +eib
cos4(x) cos4(x)
λcos(βx)µsin(γx)
cos4(x) = eix +e−ix
2
4
=1
24e4ix +4 1e3ixe−ix +4 2e2ixe−2ix +4 3eixe−3ix +e−4ix
=1
16 e4ix + 4e2ix + 6 + 4e−2ix +e−4ix =1 16 (e4ix +e−4ix) + 4(e2ix +e−2ix)+6
=1
16 (2 cos(4x) + 8 cos(2x) + 6) = 1 8cos(4x) + 1 2cos(2x) + 3 8
cos3(x) sin(x)
cos3(x) sin(x) = eix +e−ix
2
3eix −e−ix
2i=1 16ie3ix + 3eix + 3e−ix +e−3ix (eix −e−ix)
=1
16i(e4ix −e−4ix) + 2(e2ix −e−2ix) = 1 16i(2isin(4x)+4isin(2x)) = 1 8sin(4x) + 1 4sin(2x)

x7→ eix C∞Rx7→ eu(x)
u(x) = ix u0(x) = i
∀x∈R, ϕ(x) = eix,∀x∈R, ϕ0(x) = ieix
cos ∀x∈R,cos(x) = eix+e−ix
2C∞R
∀x∈R,cos0(x) = −sin(x)
cos C∞C∞R
∀x∈R,cos0(x) = ieix + (−i)e−ix
2=i 2(eix −e−ix) = i 2(2isin(x)) = i2sin(x) = −sin(x)
cos 0
cos(x)=1−x2
2! +x4 4! −x6 6! +x8 8! +··· +(−1)nx2n (2n)! +o
x→0x2n
cos C∞]−1,1[
DL 0
n>0eix =
2n
k=0
(ix)k
k!+o
x→0x2ne−ix =
2n
k=0
(ix)k
k!+o
x→0x2n)
eix +e−ix =
2n
k=0
(ik+ (−i)k)xk
k!+o
x→0x2n
k(−i)k=−ikk
k k = 2j ik+ (−i)k=i2j+ (−i)2j= (i2)j+ ((−i)2)j= (−1)j+ (−1)j= 2(−1)j
eix +e−ix =
n
j=0
2(−1)jx2j
(2j)! +o
x→0x2n
cos(x) 0
DL cos 0
ex1 + x+x2
2! +x3 3! +x4 4! +x5 5! +x6 6! +···
1 + x2
2! +x4 4! +x6 6! +···
1−x2
2! +x4 4! −x6 6! +···
cos(x)=1−x2
2+o
x→0x2cos(x)−1∼ x→0−x2
2

sin ∀x∈R,sin(x) = eix−e−ix
2iC∞R
∀x∈R,sin0(x) = cos(x)
sin C∞C∞R
∀x∈R,sin0(x) = ieix −(−i)e−ix
2i=i 2i(eix +e−ix) = 1 2(2 cos(x)) = cos(x)
sin 0
sin(x) = x−x3
2! +x5 5! −x7 7! +x9 9! +··· +(−1)nx2n+1 (2n+ 1)! +o
x→0x2n+1
DL sin 0
ex1 + x+x2
2! +x3 3! +x4 4! +x5 5! +x6 6! +···x+x3
3! +x5 5! +x7 7! +···
x−x3
3! +x5 5! −x7 7! +···
sin(x) = x+o x2sin(x)∼
x→0x
tan C∞
∀x∈R\π
2+kπ, k ∈Z,tan0(x) = 1 cos2(x)= 1 + tan2(x)
sin cos C∞Rtan C∞
C∞D=R\π
2+kπ, k ∈Z
∀x∈D, tan0(x) = sin0(x) cos(x)−sin(x) cos0(x)
(cos(x))2=cos2(x) + sin2(x) cos2(x)=
1
cos2(x)
1 + sin2(x)
cos2(x)= 1 + tan2(x)
sin(x)∼
x→0xcos(x)∼ x→01 tan(x)∼ x→0x
tan C∞]−π/2, π/2[ DL 0
xn
DL sin(x) cos(x)