Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(9+5i\\ r = \sqrt{9^2+5^2} = \sqrt{106} \\ \alpha = tan^{-1}(\frac{5}{9}) \Leftrightarrow \alpha =29^\circ 3' 16{,}6"\text{ of } \alpha = 209^\circ 3' 16{,}6"\\9+5i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 29^\circ 3' 16{,}6"\)
2. \(4+2i\\ r = \sqrt{4^2+2^2} = \sqrt{20} \\ \alpha = tan^{-1}(\frac{2}{4}) \Leftrightarrow \alpha =26^\circ 33' 54{,}2"\text{ of } \alpha = 206^\circ 33' 54{,}2"\\4+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 26^\circ 33' 54{,}2"\)
3. \(-10+i\\ r = \sqrt{(-10)^2+1^2} = \sqrt{101} \\ \alpha = tan^{-1}(\frac{1}{-10}) \Leftrightarrow \alpha =174^\circ 17' 21{,}9"\text{ of } \alpha = 354^\circ 17' 21{,}9"\\-10+i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 174^\circ 17' 21{,}9"\)
4. \(-4-6i\\ r = \sqrt{(-4)^2+(-6)^2} = \sqrt{52} \\ \alpha = tan^{-1}(\frac{-6}{-4}) \Leftrightarrow \alpha =56^\circ 18' 35{,}8"\text{ of } \alpha = 236^\circ 18' 35{,}8"\\-4-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 236^\circ 18' 35{,}8"\)
5. \(-2+10i\\ r = \sqrt{(-2)^2+10^2} = \sqrt{104} \\ \alpha = tan^{-1}(\frac{10}{-2}) \Leftrightarrow \alpha =101^\circ 18' 35{,}8"\text{ of } \alpha = 281^\circ 18' 35{,}8"\\-2+10i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 101^\circ 18' 35{,}8"\)
6. \(-2i\\ \text{ Dit complex getal ligt op het negatief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }2\\\alpha = 270 ^\circ \\\)
7. \(-8+5i\\ r = \sqrt{(-8)^2+5^2} = \sqrt{89} \\ \alpha = tan^{-1}(\frac{5}{-8}) \Leftrightarrow \alpha =147^\circ 59' 40{,}6"\text{ of } \alpha = 327^\circ 59' 40{,}6"\\-8+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 147^\circ 59' 40{,}6"\)
8. \(4-5i\\ r = \sqrt{4^2+(-5)^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{-5}{4}) \Leftrightarrow \alpha =128^\circ 39' 35{,}3"\text{ of } \alpha = 308^\circ 39' 35{,}3"\\4-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 308^\circ 39' 35{,}3"\)
9. \(9+4i\\ r = \sqrt{9^2+4^2} = \sqrt{97} \\ \alpha = tan^{-1}(\frac{4}{9}) \Leftrightarrow \alpha =23^\circ 57' 45"\text{ of } \alpha = 203^\circ 57' 45"\\9+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 23^\circ 57' 45"\)
10. \(6-6i\\ r = \sqrt{6^2+(-6)^2} = \sqrt{72} \\ \alpha = tan^{-1}(\frac{-6}{6}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\6-6i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 315^\circ \)
11. \(-8-9i\\ r = \sqrt{(-8)^2+(-9)^2} = \sqrt{145} \\ \alpha = tan^{-1}(\frac{-9}{-8}) \Leftrightarrow \alpha =48^\circ 21' 59{,}3"\text{ of } \alpha = 228^\circ 21' 59{,}3"\\-8-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 228^\circ 21' 59{,}3"\)
12. \(6+2i\\ r = \sqrt{6^2+2^2} = \sqrt{40} \\ \alpha = tan^{-1}(\frac{2}{6}) \Leftrightarrow \alpha =18^\circ 26' 5{,}8"\text{ of } \alpha = 198^\circ 26' 5{,}8"\\6+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 18^\circ 26' 5{,}8"\)