Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-9-4i\\ r = \sqrt{(-9)^2+(-4)^2} = \sqrt{97} \\ \alpha = tan^{-1}(\frac{-4}{-9}) \Leftrightarrow \alpha =23^\circ 57' 45"\text{ of } \alpha = 203^\circ 57' 45"\\-9-4i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 203^\circ 57' 45"\)
2. \(3-2i\\ r = \sqrt{3^2+(-2)^2} = \sqrt{13} \\ \alpha = tan^{-1}(\frac{-2}{3}) \Leftrightarrow \alpha =146^\circ 18' 35{,}8"\text{ of } \alpha = 326^\circ 18' 35{,}8"\\3-2i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 326^\circ 18' 35{,}8"\)
3. \(-6+10i\\ r = \sqrt{(-6)^2+10^2} = \sqrt{136} \\ \alpha = tan^{-1}(\frac{10}{-6}) \Leftrightarrow \alpha =120^\circ 57' 49{,}5"\text{ of } \alpha = 300^\circ 57' 49{,}5"\\-6+10i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 120^\circ 57' 49{,}5"\)
4. \(-6+3i\\ r = \sqrt{(-6)^2+3^2} = \sqrt{45} \\ \alpha = tan^{-1}(\frac{3}{-6}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\-6+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 153^\circ 26' 5{,}8"\)
5. \(1-10i\\ r = \sqrt{1^2+(-10)^2} = \sqrt{101} \\ \alpha = tan^{-1}(\frac{-10}{1}) \Leftrightarrow \alpha =95^\circ 42' 38{,}1"\text{ of } \alpha = 275^\circ 42' 38{,}1"\\1-10i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 275^\circ 42' 38{,}1"\)
6. \(9i\\ \text{ Dit complex getal ligt op het positief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }9\\\alpha = 90 ^\circ \\\)
7. \(7-9i\\ r = \sqrt{7^2+(-9)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-9}{7}) \Leftrightarrow \alpha =127^\circ 52' 29{,}9"\text{ of } \alpha = 307^\circ 52' 29{,}9"\\7-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 307^\circ 52' 29{,}9"\)
8. \(-8+7i\\ r = \sqrt{(-8)^2+7^2} = \sqrt{113} \\ \alpha = tan^{-1}(\frac{7}{-8}) \Leftrightarrow \alpha =138^\circ 48' 50{,}7"\text{ of } \alpha = 318^\circ 48' 50{,}7"\\-8+7i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 138^\circ 48' 50{,}7"\)
9. \(-5+9i\\ r = \sqrt{(-5)^2+9^2} = \sqrt{106} \\ \alpha = tan^{-1}(\frac{9}{-5}) \Leftrightarrow \alpha =119^\circ 3' 16{,}6"\text{ of } \alpha = 299^\circ 3' 16{,}6"\\-5+9i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 119^\circ 3' 16{,}6"\)
10. \(-7-3i\\ r = \sqrt{(-7)^2+(-3)^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{-3}{-7}) \Leftrightarrow \alpha =23^\circ 11' 54{,}9"\text{ of } \alpha = 203^\circ 11' 54{,}9"\\-7-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 203^\circ 11' 54{,}9"\)
11. \(-8-10i\\ r = \sqrt{(-8)^2+(-10)^2} = \sqrt{164} \\ \alpha = tan^{-1}(\frac{-10}{-8}) \Leftrightarrow \alpha =51^\circ 20' 24{,}7"\text{ of } \alpha = 231^\circ 20' 24{,}7"\\-8-10i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 231^\circ 20' 24{,}7"\)
12. \(-10+2i\\ r = \sqrt{(-10)^2+2^2} = \sqrt{104} \\ \alpha = tan^{-1}(\frac{2}{-10}) \Leftrightarrow \alpha =168^\circ 41' 24{,}2"\text{ of } \alpha = 348^\circ 41' 24{,}2"\\-10+2i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 168^\circ 41' 24{,}2"\)