Formelsamling till kursen Grundläggande matematik

Linnéuniversitetet
Institutionen för datavetenskap, fysik och matematik
Formelsamling till kursen Grundläggande matematik
Trigonometriska formler
sin(−𝑥) = − sin 𝑥
sin(𝜋 − 𝑥) = sin 𝑥
sin(𝜋/2 − 𝑥) = cos 𝑥
sin(𝑥 + 𝑦) = sin 𝑥 cos 𝑦 + cos 𝑥 sin 𝑦
sin 𝑥
cos 𝑥
cos(−𝑥) = cos 𝑥
cos(𝜋 − 𝑥) = − cos 𝑥
cos(𝜋/2 − 𝑥) = sin 𝑥
cos(𝑥 + 𝑦) = cos 𝑥 cos 𝑦 − sin 𝑥 sin 𝑦
sin 2𝑥 = 2 sin 𝑥 cos 𝑥
1 − cos 2𝑥
sin2 𝑥 =
2
cos 2𝑥 = 2 cos2 𝑥 − 1 = 1 − 2 sin2 𝑥 = cos2 𝑥 − sin2 𝑥
1 + cos 2𝑥
cos2 𝑥 =
2
sin2 𝑥 + cos2 𝑥 = 1
tan 𝑥 =
𝑥
0
cos 𝑥
1
sin 𝑥
0
𝜋
6
√
3
2
1
2
𝜋
4
𝜋
3
1
√
2
1
√
2
1
√2
3
2
𝜋
2
𝜋
0
−1
1
0
Potens– och logaritmlagar
Om
𝑎, 𝑏 > 0 så gäller:
𝑎0 = 1
𝑎𝑥+𝑦 = 𝑎𝑥 ⋅ 𝑎𝑦
(𝑎𝑥 )𝑦 = 𝑎𝑥𝑦
(𝑎 ⋅ 𝑏)𝑥 = 𝑎𝑥 ⋅ 𝑏𝑥
𝑎𝑥
𝑎𝑥−𝑦 = 𝑦
( 𝑎 )𝑥 𝑎𝑎𝑥
= 𝑥
𝑏
𝑏
𝑎 > 0, 𝑎 ∕= 1 så gäller:
𝑎
𝑎
log 1 = 0
log(𝑥 ⋅ 𝑦) =𝑎 log 𝑥 +𝑎 log 𝑦
𝑥
𝑎
𝑎
log( ) =𝑎 log 𝑥 −𝑎 log 𝑦
log(𝑥𝑦 ) = 𝑦 𝑎 log 𝑥
𝑦
Fakulteter och binomialkoeﬃcienter
För varje heltal 𝑛 = 1, 2, 3, . . . är 𝑛! = 𝑛(𝑛 − 1) ⋅ ⋅ ⋅ 3 ⋅ 2 ⋅ 1. För 𝑛 = 0 är 0! = 1.
( )
𝑛!
För alla heltal 𝑛, 𝑘 med 0 ≤ 𝑘 ≤ 𝑛 är 𝑛𝑘 = 𝑘!(𝑛−𝑘)!
Om
Binomialteoremet
För varje naturligt tal 𝑛 gäller
( )
( )
( )
( )
𝑛 ( )
∑
𝑛 𝑛−𝑘 𝑘
𝑛 𝑛 0
𝑛 𝑛−1 1
𝑛 𝑛−2 2
𝑛 0 𝑛
𝑛
(𝑥 + 𝑦) =
𝑥 𝑦 =
𝑥 𝑦 +
𝑥 𝑦 +
𝑥 𝑦 + ⋅⋅⋅ +
𝑥𝑦
𝑘
0
1
2
𝑛
𝑘=0
Polär form och potensform
Varje komplext tal 𝑧 = 𝑥 + 𝑖𝑦 kan skrivas 𝑧 = 𝑟𝑒𝑖𝜙 = 𝑟(cos 𝜙 + 𝑖 sin 𝜙).