Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-1-6i\\ r = \sqrt{(-1)^2+(-6)^2} = \sqrt{37} \\ \alpha = tan^{-1}(\frac{-6}{-1}) \Leftrightarrow \alpha =80^\circ 32' 15{,}6"\text{ of } \alpha = 260^\circ 32' 15{,}6"\\-1-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 260^\circ 32' 15{,}6"\)
2. \(7-9i\\ r = \sqrt{7^2+(-9)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-9}{7}) \Leftrightarrow \alpha =127^\circ 52' 29{,}9"\text{ of } \alpha = 307^\circ 52' 29{,}9"\\7-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 307^\circ 52' 29{,}9"\)
3. \(-5+5i\\ r = \sqrt{(-5)^2+5^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{5}{-5}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-5+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)
4. \(-8-2i\\ r = \sqrt{(-8)^2+(-2)^2} = \sqrt{68} \\ \alpha = tan^{-1}(\frac{-2}{-8}) \Leftrightarrow \alpha =14^\circ 2' 10{,}5"\text{ of } \alpha = 194^\circ 2' 10{,}5"\\-8-2i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 194^\circ 2' 10{,}5"\)
5. \(7+7i\\ r = \sqrt{7^2+7^2} = \sqrt{98} \\ \alpha = tan^{-1}(\frac{7}{7}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\7+7i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
6. \(8+3i\\ r = \sqrt{8^2+3^2} = \sqrt{73} \\ \alpha = tan^{-1}(\frac{3}{8}) \Leftrightarrow \alpha =20^\circ 33' 21{,}8"\text{ of } \alpha = 200^\circ 33' 21{,}8"\\8+3i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 20^\circ 33' 21{,}8"\)
7. \(-1+5i\\ r = \sqrt{(-1)^2+5^2} = \sqrt{26} \\ \alpha = tan^{-1}(\frac{5}{-1}) \Leftrightarrow \alpha =101^\circ 18' 35{,}8"\text{ of } \alpha = 281^\circ 18' 35{,}8"\\-1+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 101^\circ 18' 35{,}8"\)
8. \(6\\ \text{ Dit complex getal ligt op het positief gedeelte van de x-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }6\\\alpha = 0 ^\circ \\\)
9. \(3+6i\\ r = \sqrt{3^2+6^2} = \sqrt{45} \\ \alpha = tan^{-1}(\frac{6}{3}) \Leftrightarrow \alpha =63^\circ 26' 5{,}8"\text{ of } \alpha = 243^\circ 26' 5{,}8"\\3+6i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 63^\circ 26' 5{,}8"\)
10. \(9-i\\ r = \sqrt{9^2+(-1)^2} = \sqrt{82} \\ \alpha = tan^{-1}(\frac{-1}{9}) \Leftrightarrow \alpha =173^\circ 39' 35{,}3"\text{ of } \alpha = 353^\circ 39' 35{,}3"\\9-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 353^\circ 39' 35{,}3"\)
11. \(-10-9i\\ r = \sqrt{(-10)^2+(-9)^2} = \sqrt{181} \\ \alpha = tan^{-1}(\frac{-9}{-10}) \Leftrightarrow \alpha =41^\circ 59' 14"\text{ of } \alpha = 221^\circ 59' 14"\\-10-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 221^\circ 59' 14"\)
12. \(-1+10i\\ r = \sqrt{(-1)^2+10^2} = \sqrt{101} \\ \alpha = tan^{-1}(\frac{10}{-1}) \Leftrightarrow \alpha =95^\circ 42' 38{,}1"\text{ of } \alpha = 275^\circ 42' 38{,}1"\\-1+10i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 95^\circ 42' 38{,}1"\)