Bepaal modulus en argument

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Bepaal modulus en argument

1. \(-10-6i\)
2. \(7-9i\)
3. \(3+5i\)
4. \(-2+2i\)
5. \(4i\)
6. \(1-3i\)
7. \(9+6i\)
8. \(5-6i\)
9. \(-8-10i\)
10. \(-5-7i\)
11. \(6-5i\)
12. \(4+2i\)

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Bepaal modulus en argument

Verbetersleutel

1. \(-10-6i\\ r = \sqrt{(-10)^2+(-6)^2} = \sqrt{136} \\ \alpha = tan^{-1}(\frac{-6}{-10}) \Leftrightarrow \alpha =30^\circ 57' 49{,}5"\text{ of } \alpha = 210^\circ 57' 49{,}5"\\-10-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 210^\circ 57' 49{,}5"\)
2. \(7-9i\\ r = \sqrt{7^2+(-9)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-9}{7}) \Leftrightarrow \alpha =127^\circ 52' 29{,}9"\text{ of } \alpha = 307^\circ 52' 29{,}9"\\7-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 307^\circ 52' 29{,}9"\)
3. \(3+5i\\ r = \sqrt{3^2+5^2} = \sqrt{34} \\ \alpha = tan^{-1}(\frac{5}{3}) \Leftrightarrow \alpha =59^\circ 2' 10{,}5"\text{ of } \alpha = 239^\circ 2' 10{,}5"\\3+5i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 59^\circ 2' 10{,}5"\)
4. \(-2+2i\\ r = \sqrt{(-2)^2+2^2} = \sqrt{8} \\ \alpha = tan^{-1}(\frac{2}{-2}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-2+2i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)
5. \(4i\\ \text{ Dit complex getal ligt op het positief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }4\\\alpha = 90 ^\circ \\\)
6. \(1-3i\\ r = \sqrt{1^2+(-3)^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{-3}{1}) \Leftrightarrow \alpha =108^\circ 26' 5{,}8"\text{ of } \alpha = 288^\circ 26' 5{,}8"\\1-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 288^\circ 26' 5{,}8"\)
7. \(9+6i\\ r = \sqrt{9^2+6^2} = \sqrt{117} \\ \alpha = tan^{-1}(\frac{6}{9}) \Leftrightarrow \alpha =33^\circ 41' 24{,}2"\text{ of } \alpha = 213^\circ 41' 24{,}2"\\9+6i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 33^\circ 41' 24{,}2"\)
8. \(5-6i\\ r = \sqrt{5^2+(-6)^2} = \sqrt{61} \\ \alpha = tan^{-1}(\frac{-6}{5}) \Leftrightarrow \alpha =129^\circ 48' 20{,}1"\text{ of } \alpha = 309^\circ 48' 20{,}1"\\5-6i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 309^\circ 48' 20{,}1"\)
9. \(-8-10i\\ r = \sqrt{(-8)^2+(-10)^2} = \sqrt{164} \\ \alpha = tan^{-1}(\frac{-10}{-8}) \Leftrightarrow \alpha =51^\circ 20' 24{,}7"\text{ of } \alpha = 231^\circ 20' 24{,}7"\\-8-10i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 231^\circ 20' 24{,}7"\)
10. \(-5-7i\\ r = \sqrt{(-5)^2+(-7)^2} = \sqrt{74} \\ \alpha = tan^{-1}(\frac{-7}{-5}) \Leftrightarrow \alpha =54^\circ 27' 44{,}4"\text{ of } \alpha = 234^\circ 27' 44{,}4"\\-5-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 234^\circ 27' 44{,}4"\)
11. \(6-5i\\ r = \sqrt{6^2+(-5)^2} = \sqrt{61} \\ \alpha = tan^{-1}(\frac{-5}{6}) \Leftrightarrow \alpha =140^\circ 11' 39{,}9"\text{ of } \alpha = 320^\circ 11' 39{,}9"\\6-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 320^\circ 11' 39{,}9"\)
12. \(4+2i\\ r = \sqrt{4^2+2^2} = \sqrt{20} \\ \alpha = tan^{-1}(\frac{2}{4}) \Leftrightarrow \alpha =26^\circ 33' 54{,}2"\text{ of } \alpha = 206^\circ 33' 54{,}2"\\4+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 26^\circ 33' 54{,}2"\)