Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-3+2i\\ r = \sqrt{(-3)^2+2^2} = \sqrt{13} \\ \alpha = tan^{-1}(\frac{2}{-3}) \Leftrightarrow \alpha =146^\circ 18' 35{,}8"\text{ of } \alpha = 326^\circ 18' 35{,}8"\\-3+2i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 146^\circ 18' 35{,}8"\)
2. \(7+9i\\ r = \sqrt{7^2+9^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{9}{7}) \Leftrightarrow \alpha =52^\circ 7' 30{,}1"\text{ of } \alpha = 232^\circ 7' 30{,}1"\\7+9i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 52^\circ 7' 30{,}1"\)
3. \(4+8i\\ r = \sqrt{4^2+8^2} = \sqrt{80} \\ \alpha = tan^{-1}(\frac{8}{4}) \Leftrightarrow \alpha =63^\circ 26' 5{,}8"\text{ of } \alpha = 243^\circ 26' 5{,}8"\\4+8i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 63^\circ 26' 5{,}8"\)
4. \(5-i\\ r = \sqrt{5^2+(-1)^2} = \sqrt{26} \\ \alpha = tan^{-1}(\frac{-1}{5}) \Leftrightarrow \alpha =168^\circ 41' 24{,}2"\text{ of } \alpha = 348^\circ 41' 24{,}2"\\5-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 348^\circ 41' 24{,}2"\)
5. \(2-i\\ r = \sqrt{2^2+(-1)^2} = \sqrt{5} \\ \alpha = tan^{-1}(\frac{-1}{2}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\2-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 333^\circ 26' 5{,}8"\)
6. \(3+6i\\ r = \sqrt{3^2+6^2} = \sqrt{45} \\ \alpha = tan^{-1}(\frac{6}{3}) \Leftrightarrow \alpha =63^\circ 26' 5{,}8"\text{ of } \alpha = 243^\circ 26' 5{,}8"\\3+6i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 63^\circ 26' 5{,}8"\)
7. \(-5i\\ \text{ Dit complex getal ligt op het negatief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }5\\\alpha = 270 ^\circ \\\)
8. \(-4-5i\\ r = \sqrt{(-4)^2+(-5)^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{-5}{-4}) \Leftrightarrow \alpha =51^\circ 20' 24{,}7"\text{ of } \alpha = 231^\circ 20' 24{,}7"\\-4-5i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 231^\circ 20' 24{,}7"\)
9. \(9+5i\\ r = \sqrt{9^2+5^2} = \sqrt{106} \\ \alpha = tan^{-1}(\frac{5}{9}) \Leftrightarrow \alpha =29^\circ 3' 16{,}6"\text{ of } \alpha = 209^\circ 3' 16{,}6"\\9+5i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 29^\circ 3' 16{,}6"\)
10. \(6-3i\\ r = \sqrt{6^2+(-3)^2} = \sqrt{45} \\ \alpha = tan^{-1}(\frac{-3}{6}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\6-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 333^\circ 26' 5{,}8"\)
11. \(-2+i\\ r = \sqrt{(-2)^2+1^2} = \sqrt{5} \\ \alpha = tan^{-1}(\frac{1}{-2}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\-2+i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 153^\circ 26' 5{,}8"\)
12. \(-10-3i\\ r = \sqrt{(-10)^2+(-3)^2} = \sqrt{109} \\ \alpha = tan^{-1}(\frac{-3}{-10}) \Leftrightarrow \alpha =16^\circ 41' 57{,}3"\text{ of } \alpha = 196^\circ 41' 57{,}3"\\-10-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 196^\circ 41' 57{,}3"\)