Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-5+5i\\ r = \sqrt{(-5)^2+5^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{5}{-5}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-5+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)
2. \(-7-8i\\ r = \sqrt{(-7)^2+(-8)^2} = \sqrt{113} \\ \alpha = tan^{-1}(\frac{-8}{-7}) \Leftrightarrow \alpha =48^\circ 48' 50{,}7"\text{ of } \alpha = 228^\circ 48' 50{,}7"\\-7-8i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 228^\circ 48' 50{,}7"\)
3. \(5i\\ \text{ Dit complex getal ligt op het positief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }5\\\alpha = 90 ^\circ \\\)
4. \(-7+3i\\ r = \sqrt{(-7)^2+3^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{3}{-7}) \Leftrightarrow \alpha =156^\circ 48' 5{,}1"\text{ of } \alpha = 336^\circ 48' 5{,}1"\\-7+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 156^\circ 48' 5{,}1"\)
5. \(7-9i\\ r = \sqrt{7^2+(-9)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-9}{7}) \Leftrightarrow \alpha =127^\circ 52' 29{,}9"\text{ of } \alpha = 307^\circ 52' 29{,}9"\\7-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 307^\circ 52' 29{,}9"\)
6. \(-9+7i\\ r = \sqrt{(-9)^2+7^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{7}{-9}) \Leftrightarrow \alpha =142^\circ 7' 30{,}1"\text{ of } \alpha = 322^\circ 7' 30{,}1"\\-9+7i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 142^\circ 7' 30{,}1"\)
7. \(-4+5i\\ r = \sqrt{(-4)^2+5^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{5}{-4}) \Leftrightarrow \alpha =128^\circ 39' 35{,}3"\text{ of } \alpha = 308^\circ 39' 35{,}3"\\-4+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 128^\circ 39' 35{,}3"\)
8. \(8-10i\\ r = \sqrt{8^2+(-10)^2} = \sqrt{164} \\ \alpha = tan^{-1}(\frac{-10}{8}) \Leftrightarrow \alpha =128^\circ 39' 35{,}3"\text{ of } \alpha = 308^\circ 39' 35{,}3"\\8-10i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 308^\circ 39' 35{,}3"\)
9. \(-8+i\\ r = \sqrt{(-8)^2+1^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{1}{-8}) \Leftrightarrow \alpha =172^\circ 52' 29{,}9"\text{ of } \alpha = 352^\circ 52' 29{,}9"\\-8+i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 172^\circ 52' 29{,}9"\)
10. \(-5-i\\ r = \sqrt{(-5)^2+(-1)^2} = \sqrt{26} \\ \alpha = tan^{-1}(\frac{-1}{-5}) \Leftrightarrow \alpha =11^\circ 18' 35{,}8"\text{ of } \alpha = 191^\circ 18' 35{,}8"\\-5-i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 191^\circ 18' 35{,}8"\)
11. \(8+9i\\ r = \sqrt{8^2+9^2} = \sqrt{145} \\ \alpha = tan^{-1}(\frac{9}{8}) \Leftrightarrow \alpha =48^\circ 21' 59{,}3"\text{ of } \alpha = 228^\circ 21' 59{,}3"\\8+9i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 48^\circ 21' 59{,}3"\)
12. \(-7+7i\\ r = \sqrt{(-7)^2+7^2} = \sqrt{98} \\ \alpha = tan^{-1}(\frac{7}{-7}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-7+7i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)