Formules van de goniometrie - overzicht

Bijlage A
Formules van de goniometrie - overzicht
def
tan α =
def
sec α =
sin α
cos α
cot α =
1
cos α
cosec α =
def
cos α
sin α
def
1
sin α
sin2 α + cos2 α = 1
1 + tan2 α =
1
cos2 α
1 + cot2 α = −
sin(α ± β) = sin α cos β ± cos α sin β
tan(α ± β) =
cos(α ± β) = cos α cos β ∓ sin α sin β
tan α ± tan β
1 ∓ tan α tan β
cot(α ± β) = −
2 tan α
1 − tan2 α
sin α =
2t
1 + t2
cos α =
1 − t2
1 + t2
tan α =
2t
1 − t2
sin2 α =
1 − cos(2α)
2
cot(2α) = −
met t = tan
sin a + sin b = 2 sin
a+b
2
2
a+b
2
1 − cot2 α
2 cot α
α
cos2 α =
cos a + cos b = 2 cos
1 ∓ cot α cot β
cot α ± cot β
cos(2α) = cos2 α − sin2 α
sin(2α) = 2 sin α cos α
tan(2α) =
1
sin2 α
cos
a−b
2
cos
1 + cos(2α)
2
a−b
2
sin a − sin b = 2 sin
a−b
2
cos a − cos b = −2 sin
a−b
2
cos
a+b
2
sin
a+b
2
2 sin p sin q = sin(p + q) + sin(p − q)
2 cos p sin q = sin(p + q) − sin(p − q)
2 cos p cos q = cos(p + q) + cos(p − q)
−2 sin p sin q = cos(p + q) − cos(p − q)
1
Bijlage B
Afgeleiden van basisfuncties - overzicht
0
(f + g)0 (x) = f 0 (x) + g 0 (x)
( + 4) = 0 + 40
(f · g)0 (x) = f 0 (x) · g(x) + f (x) · g 0 (x)
( · 4) = 0 · 4 + · 40
0
(f r ) (x) = rf r−1 (x) · f 0 (x)
0
0
(r ) = rr−1 · 0
(r ∈ R)
0
f
f 0 (x) · g(x) − f (x) · g 0 (x)
(x) =
g
g(x)2
f (x) = c (c ∈ R)
f 0 (x) = 0
f (x) = x
f 0 (x) = 1
f (x) = xr
(r ∈ R)
f (x) = ex
f (x) = ax
(a ∈ R+
0 \ {1})
f 0 (x) = rxr−1
4
0
=
0 · 4 − · 40
42
0
(r ) = r · r−1 · 0
0
= e · 0
0
= a ln a · 0
f 0 (x) = ex
e
f 0 (x) = ax ln a
a
f (x) = ln x
f 0 (x) =
1
x
(ln ) =
f (x) = a log x
f 0 (x) =
1
x ln a
( a log ) =
f (x) = sin x
f 0 (x) = cos x
(sin ) = cos · 0
f (x) = cos x
f 0 (x) = − sin x
(cos ) = − sin · 0
f (x) = tan x
f 0 (x) =
1
cos2 x
(tan ) =
f (x) = cot x
f 0 (x) =
−1
sin2 x
(cot ) =
f (x) = Arcsin x
f 0 (x) = √
1
1 − x2
0
(Arcsin ) = √
f (x) = Arccos x
f 0 (x) = √
−1
1 − x2
0
(Arccos ) = √
f (x) = Arctan x
f 0 (x) =
1
1 + x2
(Arctan ) =
f (x) = Arccot x
f 0 (x) =
−1
1 + x2
(Arccot ) =
2
0
1
· 0
0
1
· 0
ln a
0
0
1
· 0
cos2 0
0
−1
· 0
sin2 0
0
1
· 0
1 − 2
−1
· 0
1 − 2
1
· 0
1 + 2
−1
· 0
1 + 2
Bijlage C
Integralen van basisfuncties - overzicht
Z
Z
sin x = − cos x + c
0 dx = c
Z
Z
dx = x + c
Z
Z
Z
xr dx =
cos x = sin x + c
1
xr+1 + c (r ∈ R \ {1})
r+1
ex dx = ex + c
ax dx =
1 x
a
ln a
1
dx = ln |x| + c
x
Z
1
1
dx = √ Arctan
k + x2
k
Z
√
1
dx = tan x + c
cos2 x
Z
1
dx = − cot x + c
sin2 x
Z
(a ∈ R+
0 \ {1})
Z
Z
Z
x
√
k
Z
+ c (k > 0)
Z
p
1
dx = ln x + k + x2 + c
k + x2
√
1
dx = Arcsin x + c = − Arccos x + c
1 − x2
1
dx = Arctan x + c = − Arccot x + c
1 + x2
√
k + x
1
1
dx = √ ln √
+ c (k > 0)
k − x2
2 k k − x
√
1
dx = Arcsin
k − x2
x
√
k
+c
k>0
Z p
p
1 p
k k + x2 dx = x k + x2 + ln x + k + x2 + c
2
2
Z p
1 p
k
x
k − x2 dx = x k − x2 + Arcsin √
+ c (k > 0)
2
2
k
Z
x 1
dx = ln tan
+c
sin x
2
Z
1
x + π/2 dx = ln tan
+c
cos x
2
Z
1
n−1
sin x dx = − sinn−1 x cos x +
n
n
Z
1
n−1
cos x dx = cosn−1 x sin x +
n
n
Z
1
1
sin x
n−2
dx =
+
cosn x
n − 1 cosn−1 x n − 1
Z
Z
lnn x dx = x lnn x − n lnn−1 x dx
Z
n
Z
n−2
sin
x dx
Z
1
1
cos x
n−2
1
dx
=
−
+
dx
sinn x
n − 1 sinn−1 x n − 1
sinn−2 x
Z
Z
1
n
xn eax dx = xn eax −
xn−1 eax dx (a 6= 0)
a
a
Z
Z
3
n
cosn−2 x dx
1
dx
cosn−2 x