Summary
WISKUNDE ALL-IN SAMENVATTING 3E GRAAD
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Samenvatting van alle leerstof gezien in de 3e graad.
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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
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