Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(6-5i\\ r = \sqrt{6^2+(-5)^2} = \sqrt{61} \\ \alpha = tan^{-1}(\frac{-5}{6}) \Leftrightarrow \alpha =140^\circ 11' 39{,}9"\text{ of } \alpha = 320^\circ 11' 39{,}9"\\6-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 320^\circ 11' 39{,}9"\)
2. \(-7-10i\\ r = \sqrt{(-7)^2+(-10)^2} = \sqrt{149} \\ \alpha = tan^{-1}(\frac{-10}{-7}) \Leftrightarrow \alpha =55^\circ 0' 28{,}7"\text{ of } \alpha = 235^\circ 0' 28{,}7"\\-7-10i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 235^\circ 0' 28{,}7"\)
3. \(-10-3i\\ r = \sqrt{(-10)^2+(-3)^2} = \sqrt{109} \\ \alpha = tan^{-1}(\frac{-3}{-10}) \Leftrightarrow \alpha =16^\circ 41' 57{,}3"\text{ of } \alpha = 196^\circ 41' 57{,}3"\\-10-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 196^\circ 41' 57{,}3"\)
4. \(-2\\ \text{ Dit complex getal ligt op het negatief gedeelte van de x-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }2\\\alpha = 180 ^\circ \\\)
5. \(-2+5i\\ r = \sqrt{(-2)^2+5^2} = \sqrt{29} \\ \alpha = tan^{-1}(\frac{5}{-2}) \Leftrightarrow \alpha =111^\circ 48' 5{,}1"\text{ of } \alpha = 291^\circ 48' 5{,}1"\\-2+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 111^\circ 48' 5{,}1"\)
6. \(4-4i\\ r = \sqrt{4^2+(-4)^2} = \sqrt{32} \\ \alpha = tan^{-1}(\frac{-4}{4}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\4-4i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 315^\circ \)
7. \(2-9i\\ r = \sqrt{2^2+(-9)^2} = \sqrt{85} \\ \alpha = tan^{-1}(\frac{-9}{2}) \Leftrightarrow \alpha =102^\circ 31' 43{,}7"\text{ of } \alpha = 282^\circ 31' 43{,}7"\\2-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 282^\circ 31' 43{,}7"\)
8. \(-4+2i\\ r = \sqrt{(-4)^2+2^2} = \sqrt{20} \\ \alpha = tan^{-1}(\frac{2}{-4}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\-4+2i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 153^\circ 26' 5{,}8"\)
9. \(8-5i\\ r = \sqrt{8^2+(-5)^2} = \sqrt{89} \\ \alpha = tan^{-1}(\frac{-5}{8}) \Leftrightarrow \alpha =147^\circ 59' 40{,}6"\text{ of } \alpha = 327^\circ 59' 40{,}6"\\8-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 327^\circ 59' 40{,}6"\)
10. \(-1+5i\\ r = \sqrt{(-1)^2+5^2} = \sqrt{26} \\ \alpha = tan^{-1}(\frac{5}{-1}) \Leftrightarrow \alpha =101^\circ 18' 35{,}8"\text{ of } \alpha = 281^\circ 18' 35{,}8"\\-1+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 101^\circ 18' 35{,}8"\)
11. \(-9+3i\\ r = \sqrt{(-9)^2+3^2} = \sqrt{90} \\ \alpha = tan^{-1}(\frac{3}{-9}) \Leftrightarrow \alpha =161^\circ 33' 54{,}2"\text{ of } \alpha = 341^\circ 33' 54{,}2"\\-9+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 161^\circ 33' 54{,}2"\)
12. \(4-9i\\ r = \sqrt{4^2+(-9)^2} = \sqrt{97} \\ \alpha = tan^{-1}(\frac{-9}{4}) \Leftrightarrow \alpha =113^\circ 57' 45"\text{ of } \alpha = 293^\circ 57' 45"\\4-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 293^\circ 57' 45"\)