Summary
WISKUNDE ALL-IN SAMENVATTING 3E GRAAD
- Course
- Institution
Samenvatting van alle leerstof gezien in de 3e graad.
[Show more]
Seller
Follow
thibautlagaert
Content preview
WiskundeMerkwaardige producten
(𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 − 𝑏𝑏) = 𝑎𝑎2 − 𝑏𝑏 2 (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎2 − 𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 ) = 𝑎𝑎3 + 𝑏𝑏 3
(𝑎𝑎 + 𝑏𝑏)2 = 𝑎𝑎2 + 2𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 (𝑎𝑎 − 𝑏𝑏)(𝑎𝑎2 + 𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 ) = 𝑎𝑎3 − 𝑏𝑏 3
(𝑎𝑎 − 𝑏𝑏)2 = 𝑎𝑎2 − 2𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 𝑛𝑛(𝑛𝑛+1)
= 1 + 2 + ⋯ + 𝑛𝑛
(𝑎𝑎 + 𝑏𝑏)3 = 𝑎𝑎3 + 3𝑎𝑎2 𝑏𝑏 + 3𝑎𝑎𝑏𝑏 2 + 𝑏𝑏 3 2
𝑛𝑛(𝑛𝑛+1)(2𝑛𝑛+1)
(𝑎𝑎 − 𝑏𝑏)3 = 𝑎𝑎3 − 3𝑎𝑎2 𝑏𝑏 + 3𝑎𝑎𝑏𝑏 2 − 𝑏𝑏 3 = 1 + 22 + ⋯ + 𝑛𝑛2
6
(𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐)2 = 𝑎𝑎2 + 𝑏𝑏 2 + 𝑐𝑐 2 + 2𝑎𝑎𝑎𝑎 + 2𝑎𝑎𝑎𝑎 + 2𝑏𝑏𝑏𝑏 𝑛𝑛(𝑛𝑛+1) 2
� � = 13 + 23 + ⋯ + 𝑛𝑛3
2
Rijen
RR MR
EXPLICIET 𝑡𝑡𝑛𝑛 = 𝑡𝑡𝑎𝑎 + (𝑛𝑛 − 𝑎𝑎)𝑣𝑣 𝑡𝑡𝑛𝑛 = 𝑡𝑡𝑎𝑎 ∗ 𝑞𝑞 𝑛𝑛−𝑎𝑎
met term a gegeven met term a gegeven
RECURSIEF 𝑡𝑡𝑛𝑛 = 𝑡𝑡𝑛𝑛−1 + 𝑣𝑣 𝑡𝑡𝑛𝑛 = 𝑡𝑡𝑛𝑛−1 ∗ 𝑞𝑞
SN 𝑡𝑡1 + 𝑡𝑡𝑛𝑛 𝑞𝑞 ≠ 1: 1−𝑞𝑞 𝑛𝑛
𝑆𝑆𝑛𝑛 = 𝑛𝑛 𝑆𝑆𝑛𝑛 = 𝑡𝑡1
1−𝑞𝑞
2 𝑞𝑞 = 1: 𝑆𝑆𝑛𝑛 = 𝑡𝑡1 𝑛𝑛
𝑡𝑡
−1 < 𝑞𝑞 < 1: 𝑆𝑆 = 1
1−𝑞𝑞
Matrices
rekenregels • determinanten: begrippen
• ℝm x n, + is een commutatieve groep • minor (v. amn): det. (m’e rij en n’e kolom weglaten)
𝑂𝑂 ∈ ℝ𝑚𝑚 𝑥𝑥 𝑛𝑛 , ∀𝐴𝐴, 𝐵𝐵, 𝐶𝐶 ∈ ℝ𝑚𝑚 𝑥𝑥 𝑛𝑛 : • cofactor Aij: (-1)i+j* minor aij
• 𝐴𝐴 + 𝐵𝐵 ∈ ℝ𝑚𝑚 𝑥𝑥 𝑛𝑛 • determinanten: eigenschappen
• (𝐴𝐴 + 𝐵𝐵) + 𝐶𝐶 = 𝐴𝐴 + (𝐵𝐵 + 𝐶𝐶) ∀𝐴𝐴, 𝐵𝐵 ∈ ℝ𝑛𝑛 𝑥𝑥 𝑛𝑛 :
• 𝐴𝐴 + 𝑂𝑂 = 𝐴𝐴 = 𝑂𝑂 + 𝐴𝐴 • det 𝐴𝐴 = det 𝐴𝐴𝑇𝑇
• ∃! − 𝐴𝐴 ∈ ℝ𝑚𝑚 𝑥𝑥 𝑛𝑛 : 𝐴𝐴 + (−𝐴𝐴) = 𝑂𝑂 = (−𝐴𝐴) + 𝐴𝐴 𝑅𝑅
𝐾𝐾12
• 𝐴𝐴 + 𝐵𝐵 = 𝐵𝐵 + 𝐴𝐴 • 𝐴𝐴 �� 𝐵𝐵: det 𝐵𝐵 = − det 𝐴𝐴
• ℝ, ℝm x n, + is een reële vectorruimte • 2 gelijke/evenredige R/K of nulrij ⇒ det 𝐴𝐴 = 0
𝑅𝑅
𝑂𝑂 ∈ ℝ𝑚𝑚 𝑥𝑥 𝑛𝑛 , ∀𝐴𝐴, 𝐵𝐵 ∈ ℝ𝑚𝑚 𝑥𝑥 𝑛𝑛 , ∀𝑟𝑟, 𝑠𝑠 ∈ ℝ: 𝐾𝐾𝑛𝑛
∗𝑡𝑡
• ℝm x n, + is een commutatieve groep • 𝐴𝐴 �⎯� 𝐵𝐵: det 𝐵𝐵 = 𝑡𝑡 ∗ det 𝐴𝐴
• r ∗ A ∈ ℝm x n • factor n R/K afzonderen: 𝑛𝑛 ∗ det 𝐴𝐴
𝑎𝑎11 𝑎𝑎12 𝑎𝑎13 𝑎𝑎11 𝑎𝑎12 𝑎𝑎13 ′ 𝑎𝑎11 𝑎𝑎12 𝑎𝑎13 + 𝑎𝑎13 ′
• r ∗ (s ∗ ℝm x n ) = (r ∗ s) ∗ ℝm x n • R/K: �𝑎𝑎21 𝑎𝑎22 𝑎𝑎23 � + �𝑎𝑎21 𝑎𝑎22 𝑎𝑎23 ′� = �𝑎𝑎21 𝑎𝑎22 𝑎𝑎23 + 𝑎𝑎33 ′�
𝑎𝑎31 𝑎𝑎32 𝑎𝑎33
• r ∗ (A + B) = r ∗ A + r ∗ B 𝑎𝑎31 𝑎𝑎32 𝑎𝑎33 ′ 𝑎𝑎31 𝑎𝑎32 𝑎𝑎33 + 𝑎𝑎33 ′
𝑅𝑅 𝑅𝑅
• 1∗A = A 𝐾𝐾𝑚𝑚
+𝑡𝑡∗𝐾𝐾
𝑛𝑛
• 𝐴𝐴 �⎯⎯⎯⎯⎯� 𝐵𝐵: det 𝐵𝐵 = det 𝐴𝐴
(𝑟𝑟 ∗ 𝐴𝐴)𝑇𝑇 = 𝑟𝑟 ∗ 𝐴𝐴𝑇𝑇
• ℝ n, ∙ is geen commutatieve groep
m x • det(𝐴𝐴 ∗ 𝐵𝐵) = det 𝐴𝐴 ∗ det 𝐵𝐵
𝑂𝑂 ∈ ℝ𝑛𝑛 𝑥𝑥 𝑛𝑛 , 𝐼𝐼𝑛𝑛 ∈ ℝ𝑛𝑛 𝑥𝑥 𝑛𝑛 , ∀𝐴𝐴, 𝐵𝐵, 𝐶𝐶 ∈ ℝ𝑛𝑛 𝑥𝑥 𝑛𝑛 , ∀𝑟𝑟 ∈ ℝ: • determinanten: inversen
• 𝐴𝐴 ∗ 𝐵𝐵 ∈ ℝ𝑛𝑛 𝑥𝑥 𝑛𝑛 ∀𝐴𝐴, 𝐵𝐵 ∈ ℝ𝑛𝑛 𝑥𝑥 𝑛𝑛 :
𝑇𝑇
• 𝐴𝐴 ∗ (𝐵𝐵 ∗ 𝐶𝐶) = (𝐴𝐴 ∗ 𝐵𝐵) ∗ 𝐶𝐶 • adjunct: 𝑎𝑎𝑎𝑎𝑎𝑎 𝐴𝐴 = �𝐴𝐴𝑖𝑖𝑖𝑖 � = �𝐴𝐴𝑗𝑗𝑗𝑗 �
• 𝐴𝐴 ∗ 𝐼𝐼𝑛𝑛 = 𝐴𝐴 = 𝐼𝐼𝑛𝑛 ∗ 𝐴𝐴 • 𝐴𝐴 ∗ (𝑎𝑎𝑎𝑎𝑎𝑎 𝐴𝐴) = (𝑎𝑎𝑎𝑎𝑎𝑎 𝐴𝐴) ∗ 𝐴𝐴 = (det 𝐴𝐴) ∗ 𝐼𝐼
• nuldelers: 𝐴𝐴 ∗ 𝐵𝐵 = 𝑂𝑂 ∨ 𝐵𝐵 ∗ 𝐴𝐴 = 𝑂𝑂 met 𝐴𝐴 ≠ 𝑂𝑂 ∧ 𝐵𝐵 ≠ 𝑂𝑂 • 𝐴𝐴 𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ⇔ det 𝐴𝐴 ≠ 0
• 𝐴𝐴 ∗ (𝐵𝐵 + 𝐶𝐶) = 𝐴𝐴𝐴𝐴 + 𝐴𝐴𝐴𝐴 en (𝐵𝐵 + 𝐶𝐶) ∗ 𝐴𝐴 = 𝐵𝐵𝐵𝐵 + 𝐶𝐶𝐶𝐶 • 𝐴𝐴 𝑖𝑖𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ⇔ det 𝐴𝐴 = 0
𝑎𝑎𝑎𝑎𝑎𝑎 𝐴𝐴
• 𝐴𝐴 ∗ 𝑂𝑂 = 𝑂𝑂 = 𝑂𝑂 ∗ 𝐴𝐴 • det 𝐴𝐴 ≠ 0 ⇒ 𝐴𝐴−1 =
det 𝐴𝐴
• (𝐴𝐴 + 𝐵𝐵)𝑇𝑇 = 𝐴𝐴𝑇𝑇 + 𝐵𝐵𝑇𝑇 • |𝐴𝐴−1 | =
1
• idempotent: 𝐴𝐴2 = 𝐴𝐴 |𝐴𝐴|
• nilpotent met index n: 𝐴𝐴𝑛𝑛 = 𝑂𝑂 • determinanten: stelsel van Cramer
|𝐴𝐴 | |𝐴𝐴2 | |𝐴𝐴3 |
• involutorisch: 𝐴𝐴2 = 𝐼𝐼 • 𝑉𝑉 = �� |𝐴𝐴|1 , |𝐴𝐴|
, |𝐴𝐴|
��
• ∀𝐴𝐴 ∈ ℝ𝑚𝑚 𝑥𝑥 𝑛𝑛 , ∀𝐵𝐵 ∈ ℝ𝑛𝑛 𝑥𝑥 𝑝𝑝 : (𝐴𝐴 ∗ 𝐵𝐵)𝑇𝑇 = 𝐵𝐵𝑇𝑇 ∗ 𝐴𝐴𝑇𝑇 • homogeen 2 x 3 stelsel:
• inverse matrices 𝑎𝑎 𝑥𝑥 + 𝑏𝑏1 𝑦𝑦 + 𝑐𝑐1 𝑧𝑧 = 0 𝑎𝑎 𝑏𝑏1
� 1 met � 1 �≠0
• A-1 is de inverse van A ⇔ 𝐴𝐴 ∗ 𝐴𝐴−1 = 𝐼𝐼 = 𝐴𝐴−1 ∗ 𝐴𝐴 𝑎𝑎2 𝑥𝑥 + 𝑏𝑏2 𝑦𝑦 + 𝑐𝑐2 𝑧𝑧 = 0 𝑎𝑎2 𝑏𝑏2
𝑏𝑏 𝑐𝑐1 𝑎𝑎1 𝑐𝑐1 𝑎𝑎1 𝑏𝑏1
• 𝐼𝐼 −1 = 𝐼𝐼 𝑉𝑉 = ��𝜑𝜑 � 1 � , −𝜑𝜑 �𝑎𝑎
𝑏𝑏2 𝑐𝑐2 2 𝑐𝑐2 � , 𝜑𝜑 �𝑎𝑎2 𝑏𝑏2
� |𝜑𝜑 ∈ ℝ��
• (𝐴𝐴−1 )−1 = 𝐴𝐴
• r(A) = n, met nxn-determinant ≠ 0
• (𝐴𝐴 ∗ 𝐵𝐵)−1 = 𝐵𝐵 −1 ∗ 𝐴𝐴−1
• determinanten: eigenwaarden & eigenvectoren
• 𝐵𝐵 = 𝐶𝐶 ⇔ 𝐴𝐴 ∗ 𝐵𝐵 = 𝐴𝐴 ∗ 𝐶𝐶
• X is een eigenvector van A met eigenwaarde λ
⇔ 𝐴𝐴𝐴𝐴 = 𝜆𝜆𝜆𝜆 ⇔ (𝐴𝐴 − 𝜆𝜆𝜆𝜆) ∗ 𝑋𝑋 = 0
met det(𝐴𝐴 − 𝜆𝜆𝜆𝜆) = 0 (karakteristieke vgl v A)
P a g i n a |1
, Goniometrie
Goniometrische getallen
0° 30° 45° 60° 90° sin 𝛼𝛼 cos 𝛼𝛼 1 1 1
𝜋𝜋 𝜋𝜋 𝜋𝜋 𝜋𝜋 tan α = cot 𝛼𝛼 = = sec 𝛼𝛼 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼 =
0 cos 𝛼𝛼 sin 𝛼𝛼 tan 𝛼𝛼 cos 𝛼𝛼 sin 𝛼𝛼
6 4 3 2
sin α
0
1 √2 √3 1
Formules
2 2 2 • cos 2 (𝛼𝛼) + sin2 (𝛼𝛼) = 1
cos α √3 √2 1
1 0 • 1 + tan2 (𝛼𝛼) = 𝑠𝑠𝑠𝑠𝑐𝑐 2 (𝛼𝛼) (cos²α ≠ 0)
2 2 2
tan α • 1 + cot 2 (𝛼𝛼) = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2 (𝛼𝛼) (sin²α ≠ 0)
√3
0 1 √3 −
3
cot α √3
− √3 1 0
3
• somformules • formules van modeSimpson
• cos(𝛼𝛼 + 𝛽𝛽) = cos(𝛼𝛼) ∗ cos(𝛽𝛽) − sin(𝛼𝛼) ∗ sin(𝛽𝛽) 𝛼𝛼+𝛽𝛽 𝛼𝛼−𝛽𝛽
• sin(𝛼𝛼) + sin(𝛽𝛽) = 2 ∗ sin � � ∗ cos � �
• cos(𝛼𝛼 − 𝛽𝛽) = cos(𝛼𝛼) ∗ cos(𝛽𝛽) + sin(𝛼𝛼) ∗ sin(𝛽𝛽) 2 2
𝛼𝛼+𝛽𝛽 𝛼𝛼−𝛽𝛽
• sin(𝛼𝛼 + 𝛽𝛽) = sin(𝛼𝛼) ∗ cos(𝛽𝛽) + cos(𝛼𝛼) ∗ sin(𝛽𝛽) • sin(𝛼𝛼) − sin(𝛽𝛽) = 2 ∗ cos � � ∗ sin � �
2 2
• sin(𝛼𝛼 − 𝛽𝛽) = sin(𝛼𝛼) ∗ cos(𝛽𝛽) − cos(𝛼𝛼) ∗ sin(𝛽𝛽) • cos(𝛼𝛼) + cos(𝛽𝛽) = 2 ∗ cos �
𝛼𝛼+𝛽𝛽
� ∗ cos �
𝛼𝛼−𝛽𝛽
�
tan(𝛼𝛼)+tan(𝛽𝛽) 2 2
• tan(𝛼𝛼 + 𝛽𝛽) = 𝛼𝛼+𝛽𝛽 𝛼𝛼−𝛽𝛽
1−tan(𝛼𝛼) tan(𝛽𝛽) • cos(𝛼𝛼) − cos(𝛽𝛽) = −2 ∗ sin � � ∗ sin � �
tan(𝛼𝛼)−tan(𝛽𝛽) 2 2
• tan(𝛼𝛼 − 𝛽𝛽) = • 2 ∗ cos(𝛼𝛼) ∗ cos(𝛽𝛽) = cos(𝛼𝛼 + 𝛽𝛽) + cos(𝛼𝛼 − 𝛽𝛽)
1+tan(𝛼𝛼) tan(𝛽𝛽)
• verdubbelingsformules • 2 ∗ sin(𝛼𝛼) ∗ sin(𝛽𝛽) = cos(𝛼𝛼 − 𝛽𝛽) − cos(𝛼𝛼 + 𝛽𝛽)
• sin(2𝛼𝛼) = 2 sin(𝛼𝛼) ∗ cos(𝛼𝛼) =
2 tan(𝛼𝛼) • 2 ∗ sin(𝛼𝛼) ∗ cos(𝛽𝛽) = sin(𝛼𝛼 + 𝛽𝛽) + sin(𝛼𝛼 − 𝛽𝛽)
1+𝑡𝑡𝑡𝑡𝑛𝑛2 (𝛼𝛼)
• 2 ∗ cos(𝛼𝛼) ∗ sin(𝛽𝛽) = sin(𝛼𝛼 + 𝛽𝛽) − sin(𝛼𝛼 − 𝛽𝛽)
• cos(2𝛼𝛼) = cos 2 (𝛼𝛼) − sin2 (𝛼𝛼) = 1 − 2 ∗ sin2 (𝛼𝛼) =
• T-formules (met t = tan (α/2))
1−𝑡𝑡𝑡𝑡𝑛𝑛2 (𝛼𝛼)
2 ∗ cos 2 (𝛼𝛼) − 1 = • sin(𝛼𝛼) =
2𝑡𝑡
1+𝑡𝑡𝑡𝑡𝑛𝑛2 (𝛼𝛼)
1+𝑡𝑡²
2∗tan(𝛼𝛼)
• tan(2𝛼𝛼) = • cos(𝛼𝛼) =
1−𝑡𝑡²
1−𝑡𝑡𝑡𝑡𝑛𝑛2 (𝛼𝛼) 1+𝑡𝑡²
• halveringsformules • tan(𝛼𝛼) =
2𝑡𝑡
1−cos(2𝛼𝛼) 1−𝑡𝑡²
• sin2 (𝛼𝛼) =
2
1+cos(2𝛼𝛼)
• cos 2 (𝛼𝛼) =
2
Goniometrische vergelijkingen
• basisvergelijkingen
• sin(𝑥𝑥) = 𝑎𝑎 ⇔ 𝑥𝑥 = 𝛼𝛼 + 𝑘𝑘2𝜋𝜋 ⋁ 𝑥𝑥 = 𝜋𝜋 − 𝛼𝛼 + 𝑘𝑘2𝜋𝜋 met k ∈ ℤ, sin α = a, a ∈ [-1, 1]
• cos(𝑥𝑥) = 𝑎𝑎 ⇔ 𝑥𝑥 = 𝛼𝛼 + 𝑘𝑘2𝜋𝜋 ⋁ 𝑥𝑥 = −𝛼𝛼 + 𝑘𝑘2𝜋𝜋 met k ∈ ℤ, cos α = a, a ∈ [-1, 1]
• tan(𝑥𝑥) = 𝑎𝑎 ⇔ 𝑥𝑥 = 𝛼𝛼 + 𝑘𝑘𝑘𝑘 met k ∈ ℤ, tan α = a
• oplossingstechnieken
• uiteenvallen in basisvergelijkingen: sin 2𝑥𝑥 + cos 𝑥𝑥 = 0 ⇔ cos 𝑥𝑥 ∗ (2 sin 𝑥𝑥 + 1) = 0
• gebruik v. eenvoudige subsituties: 2 cos 2 𝑥𝑥 − cos 𝑥𝑥 + 1 = 0 ⇔ 2𝑡𝑡 2 − 𝑡𝑡 + 1 = 0
𝑐𝑐 𝑏𝑏
• vglk v/d vorm 𝑎𝑎 ∗ sin 𝑥𝑥 + 𝑏𝑏 ∗ cos 𝑥𝑥 = 𝑐𝑐: sin(𝑥𝑥 + 𝛼𝛼) = cos 𝛼𝛼 met tan 𝛼𝛼 =
𝑎𝑎 𝑎𝑎
• homogene vgl’en: afzonderen gem. factoren; delen door cos n 𝑥𝑥 met 𝑛𝑛 = graad v. overblijvende functie
𝑡𝑡 2 −1
• symmetische vgl’en (cos met sin wisselen: zelfde f): sin 𝛼𝛼 + cos 𝛼𝛼 = 𝑡𝑡, sin 𝛼𝛼 ∗ cos 𝛼𝛼 =
2
Cyclometrische functies
𝜋𝜋 𝜋𝜋
• 𝛼𝛼 = Bgsin 𝑥𝑥 ⇔ 𝑥𝑥 = sin 𝛼𝛼 ∧ 𝛼𝛼 ∈ �− , � 𝛼𝛼 = Bgcos 𝑥𝑥 ⇔ 𝑥𝑥 = cos 𝛼𝛼 ∧ 𝛼𝛼 ∈ [0, 𝜋𝜋]
2 2
𝜋𝜋 𝜋𝜋
𝛼𝛼 = Bgtan 𝑥𝑥 ⇔ 𝑥𝑥 = tan 𝛼𝛼 ∧ 𝛼𝛼 ∈ �− , � 𝛼𝛼 = Bgcot 𝑥𝑥 ⇔ 𝑥𝑥 = cot 𝛼𝛼 ∧ 𝛼𝛼 ∈ ]0, 𝜋𝜋[
2 2
𝜋𝜋
• ∀𝑥𝑥 ∈ [−1, 1]: Bgsin 𝑥𝑥 + Bgcos 𝑥𝑥 = ↗ Bgsin x Bgcos x Bgtan x Bgcot x
2
∀𝑥𝑥 ∈ ℝ: Bgtan 𝑥𝑥 + Bgcot 𝑥𝑥 =
𝜋𝜋 𝑥𝑥 1
2 sin 𝑥𝑥 �1 − 𝑥𝑥 2
𝜋𝜋 𝜋𝜋 √1 + 𝑥𝑥 2 √1 + 𝑥𝑥 2
• Bgsin(sin 𝛼𝛼) = 𝑥𝑥 ⇔ − ≤ 𝑥𝑥 ≤ 1 𝑥𝑥
2 2
Bgcos(cos 𝛼𝛼) = 𝑥𝑥 ⇔ 0 ≤ 𝑥𝑥 ≤ 𝜋𝜋 cos �1 − 𝑥𝑥 2 𝑥𝑥
𝜋𝜋 𝜋𝜋 √1 + 𝑥𝑥 2 √1 + 𝑥𝑥 2
Bgtan(tan 𝛼𝛼) = 𝑥𝑥 ⇔ − < 𝑥𝑥 < 𝑥𝑥 √1 − 𝑥𝑥 2 1
2 2 tan 𝑥𝑥
Bgcot(cot 𝛼𝛼) = 𝑥𝑥 ⇔ 0 < 𝑥𝑥 < 𝜋𝜋 √1 − 𝑥𝑥 2 𝑥𝑥 𝑥𝑥
√1 − 𝑥𝑥 2 1 𝑥𝑥
cot 2
𝑥𝑥
𝑥𝑥 √1 − 𝑥𝑥 𝑥𝑥
Uitrekenen d.m.v. omvormen hoofdformule OF mbv driehoeken
P a g i n a |2
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller thibautlagaert. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.07. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
62890 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now