xcos(x) sin(x)
xR,cos(x) = eix =eix +eix
2xR,sin(x) = eix =eix eix
2i
cos sin R
xR,16cos(x)6116sin(x)61 cos2(x) + sin2(x)=1
[1,1]
x, y R
cos(x) = cos(y)kZ/ x =y+ 2kπ
kZ/ x =y+ 2kπ
sin(x) = sin(y)kZ/ x =y+ 2kπ
kZ/ x = (πy)+2kπ
cos(x)=0 ⇒ ∃kZ/ x =π
2+kπ sin(x)=0 ⇒ ∃kZ/ x =kπ
cos(x)=1 ⇒ ∃kZ/ x = 2kπ sin(x)=1 ⇒ ∃kZ/ x =π
2+ 2kπ
cos(x) = 1⇒ ∃kZ/ x =π+ 2kπ sin(x) = 1⇒ ∃kZ/ 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
313??
2πR
2πR
π
xR,cos(x) = cos(x) cos(x+ 2π) = cos(x)
xR,sin(x) = sin(x) sin(x+ 2π) = sin(x)
xR\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(πx) = cos(x) sin(πx) = sin(x) tan(πx) = tan(x)
cos π
2x= sin(x) sin π
2x= cos(x) tan π
2x=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 ab
2cos a+b
2,sin(a) + sin(b) = 2 cos ab
2sin a+b
2
cos(a) + cos(b) = (eia) + (eib) = (eia +eib) sin(a) + sin(b) = (eia +eib)
eia +eib =eia+b
2eiab
2+eiba
2=eia+b
22 cos ab
2
= 2 cos ab
2cos a+b
2+i2 cos ab
2sin a+b
2
eia +eib
cos4(x) cos4(x)
λcos(βx)µsin(γx)
cos4(x) = eix +eix
2
4
=1
24e4ix +4
1e3ixeix +4
2e2ixe2ix +4
3eixe3ix +e4ix
=1
16 e4ix + 4e2ix + 6 + 4e2ix +e4ix =1
16 (e4ix +e4ix) + 4(e2ix +e2ix)+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 +eix
2
3eix eix
2i=1
16ie3ix + 3eix + 3eix +e3ix (eix eix)
=1
16i(e4ix e4ix) + 2(e2ix e2ix) = 1
16i(2isin(4x)+4isin(2x)) = 1
8sin(4x) + 1
4sin(2x)
x7→ eix CRx7→ eu(x)
u(x) = ix u0(x) = i
xR, ϕ(x) = eix,xR, ϕ0(x) = ieix
cos xR,cos(x) = eix+eix
2CR
xR,cos0(x) = sin(x)
cos CCR
xR,cos0(x) = ieix + (i)eix
2=i
2(eix eix) = i
2(2isin(x)) = i2sin(x) = sin(x)
cos 0
cos(x)=1x2
2! +x4
4! x6
6! +x8
8! +··· +(1)nx2n
(2n)! +o
x0x2n
cos C]1,1[
DL 0
n>0eix =
2n
k=0
(ix)k
k!+o
x0x2neix =
2n
k=0
(ix)k
k!+o
x0x2n)
eix +eix =
2n
k=0
(ik+ (i)k)xk
k!+o
x0x2n
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 +eix =
n
j=0
2(1)jx2j
(2j)! +o
x0x2n
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! +···
1x2
2! +x4
4! x6
6! +···
cos(x)=1x2
2+o
x0x2cos(x)1
x0x2
2
sin xR,sin(x) = eixeix
2iCR
xR,sin0(x) = cos(x)
sin CCR
xR,sin0(x) = ieix (i)eix
2i=i
2i(eix +eix) = 1
2(2 cos(x)) = cos(x)
sin 0
sin(x) = xx3
2! +x5
5! x7
7! +x9
9! +··· +(1)nx2n+1
(2n+ 1)! +o
x0x2n+1
DL sin 0
ex1 + x+x2
2! +x3
3! +x4
4! +x5
5! +x6
6! +···x+x3
3! +x5
5! +x7
7! +···
xx3
3! +x5
5! x7
7! +···
sin(x) = x+o x2sin(x)
x0x
tan C
xR\π
2+kπ, k Z,tan0(x) = 1
cos2(x)= 1 + tan2(x)
sin cos CRtan C
CD=R\π
2+kπ, k Z
xD, 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)
x0xcos(x)
x01 tan(x)
x0x
tan C]π/2, π/2[ DL 0
xn
DL sin(x) cos(x)
1 / 8 100%
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