Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-1+3i\\ r = \sqrt{(-1)^2+3^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{3}{-1}) \Leftrightarrow \alpha =108^\circ 26' 5{,}8"\text{ of } \alpha = 288^\circ 26' 5{,}8"\\-1+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 108^\circ 26' 5{,}8"\)
2. \(10+2i\\ r = \sqrt{10^2+2^2} = \sqrt{104} \\ \alpha = tan^{-1}(\frac{2}{10}) \Leftrightarrow \alpha =11^\circ 18' 35{,}8"\text{ of } \alpha = 191^\circ 18' 35{,}8"\\10+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 11^\circ 18' 35{,}8"\)
3. \(4-5i\\ r = \sqrt{4^2+(-5)^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{-5}{4}) \Leftrightarrow \alpha =128^\circ 39' 35{,}3"\text{ of } \alpha = 308^\circ 39' 35{,}3"\\4-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 308^\circ 39' 35{,}3"\)
4. \(-8+6i\\ r = \sqrt{(-8)^2+6^2} = \sqrt{100} \\ \alpha = tan^{-1}(\frac{6}{-8}) \Leftrightarrow \alpha =143^\circ 7' 48{,}4"\text{ of } \alpha = 323^\circ 7' 48{,}4"\\-8+6i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 143^\circ 7' 48{,}4"\)
5. \(10+i\\ r = \sqrt{10^2+1^2} = \sqrt{101} \\ \alpha = tan^{-1}(\frac{1}{10}) \Leftrightarrow \alpha =5^\circ 42' 38{,}1"\text{ of } \alpha = 185^\circ 42' 38{,}1"\\10+i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 5^\circ 42' 38{,}1"\)
6. \(-7-5i\\ r = \sqrt{(-7)^2+(-5)^2} = \sqrt{74} \\ \alpha = tan^{-1}(\frac{-5}{-7}) \Leftrightarrow \alpha =35^\circ 32' 15{,}6"\text{ of } \alpha = 215^\circ 32' 15{,}6"\\-7-5i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 215^\circ 32' 15{,}6"\)
7. \(6+2i\\ r = \sqrt{6^2+2^2} = \sqrt{40} \\ \alpha = tan^{-1}(\frac{2}{6}) \Leftrightarrow \alpha =18^\circ 26' 5{,}8"\text{ of } \alpha = 198^\circ 26' 5{,}8"\\6+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 18^\circ 26' 5{,}8"\)
8. \(9+i\\ r = \sqrt{9^2+1^2} = \sqrt{82} \\ \alpha = tan^{-1}(\frac{1}{9}) \Leftrightarrow \alpha =6^\circ 20' 24{,}7"\text{ of } \alpha = 186^\circ 20' 24{,}7"\\9+i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 6^\circ 20' 24{,}7"\)
9. \(-7-9i\\ r = \sqrt{(-7)^2+(-9)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-9}{-7}) \Leftrightarrow \alpha =52^\circ 7' 30{,}1"\text{ of } \alpha = 232^\circ 7' 30{,}1"\\-7-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 232^\circ 7' 30{,}1"\)
10. \(-2i\\ \text{ Dit complex getal ligt op het negatief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }2\\\alpha = 270 ^\circ \\\)
11. \(7-3i\\ r = \sqrt{7^2+(-3)^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{-3}{7}) \Leftrightarrow \alpha =156^\circ 48' 5{,}1"\text{ of } \alpha = 336^\circ 48' 5{,}1"\\7-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 336^\circ 48' 5{,}1"\)
12. \(10-4i\\ r = \sqrt{10^2+(-4)^2} = \sqrt{116} \\ \alpha = tan^{-1}(\frac{-4}{10}) \Leftrightarrow \alpha =158^\circ 11' 54{,}9"\text{ of } \alpha = 338^\circ 11' 54{,}9"\\10-4i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 338^\circ 11' 54{,}9"\)