Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(3i\\ \text{ Dit complex getal ligt op het positief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }3\\\alpha = 90 ^\circ \\\)
2. \(-6+6i\\ r = \sqrt{(-6)^2+6^2} = \sqrt{72} \\ \alpha = tan^{-1}(\frac{6}{-6}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-6+6i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)
3. \(-10-2i\\ r = \sqrt{(-10)^2+(-2)^2} = \sqrt{104} \\ \alpha = tan^{-1}(\frac{-2}{-10}) \Leftrightarrow \alpha =11^\circ 18' 35{,}8"\text{ of } \alpha = 191^\circ 18' 35{,}8"\\-10-2i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 191^\circ 18' 35{,}8"\)
4. \(-5-9i\\ r = \sqrt{(-5)^2+(-9)^2} = \sqrt{106} \\ \alpha = tan^{-1}(\frac{-9}{-5}) \Leftrightarrow \alpha =60^\circ 56' 43{,}4"\text{ of } \alpha = 240^\circ 56' 43{,}4"\\-5-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 240^\circ 56' 43{,}4"\)
5. \(-9-7i\\ r = \sqrt{(-9)^2+(-7)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-7}{-9}) \Leftrightarrow \alpha =37^\circ 52' 29{,}9"\text{ of } \alpha = 217^\circ 52' 29{,}9"\\-9-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 217^\circ 52' 29{,}9"\)
6. \(-3-5i\\ r = \sqrt{(-3)^2+(-5)^2} = \sqrt{34} \\ \alpha = tan^{-1}(\frac{-5}{-3}) \Leftrightarrow \alpha =59^\circ 2' 10{,}5"\text{ of } \alpha = 239^\circ 2' 10{,}5"\\-3-5i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 239^\circ 2' 10{,}5"\)
7. \(-5+2i\\ r = \sqrt{(-5)^2+2^2} = \sqrt{29} \\ \alpha = tan^{-1}(\frac{2}{-5}) \Leftrightarrow \alpha =158^\circ 11' 54{,}9"\text{ of } \alpha = 338^\circ 11' 54{,}9"\\-5+2i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 158^\circ 11' 54{,}9"\)
8. \(-9+4i\\ r = \sqrt{(-9)^2+4^2} = \sqrt{97} \\ \alpha = tan^{-1}(\frac{4}{-9}) \Leftrightarrow \alpha =156^\circ 2' 15"\text{ of } \alpha = 336^\circ 2' 15"\\-9+4i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 156^\circ 2' 15"\)
9. \(3+8i\\ r = \sqrt{3^2+8^2} = \sqrt{73} \\ \alpha = tan^{-1}(\frac{8}{3}) \Leftrightarrow \alpha =69^\circ 26' 38{,}2"\text{ of } \alpha = 249^\circ 26' 38{,}2"\\3+8i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 69^\circ 26' 38{,}2"\)
10. \(1+4i\\ r = \sqrt{1^2+4^2} = \sqrt{17} \\ \alpha = tan^{-1}(\frac{4}{1}) \Leftrightarrow \alpha =75^\circ 57' 49{,}5"\text{ of } \alpha = 255^\circ 57' 49{,}5"\\1+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 75^\circ 57' 49{,}5"\)
11. \(-4+5i\\ r = \sqrt{(-4)^2+5^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{5}{-4}) \Leftrightarrow \alpha =128^\circ 39' 35{,}3"\text{ of } \alpha = 308^\circ 39' 35{,}3"\\-4+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 128^\circ 39' 35{,}3"\)
12. \(5-9i\\ r = \sqrt{5^2+(-9)^2} = \sqrt{106} \\ \alpha = tan^{-1}(\frac{-9}{5}) \Leftrightarrow \alpha =119^\circ 3' 16{,}6"\text{ of } \alpha = 299^\circ 3' 16{,}6"\\5-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 299^\circ 3' 16{,}6"\)