Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(6+9i\\ r = \sqrt{6^2+9^2} = \sqrt{117} \\ \alpha = tan^{-1}(\frac{9}{6}) \Leftrightarrow \alpha =56^\circ 18' 35{,}8"\text{ of } \alpha = 236^\circ 18' 35{,}8"\\6+9i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 56^\circ 18' 35{,}8"\)
2. \(4-9i\\ r = \sqrt{4^2+(-9)^2} = \sqrt{97} \\ \alpha = tan^{-1}(\frac{-9}{4}) \Leftrightarrow \alpha =113^\circ 57' 45"\text{ of } \alpha = 293^\circ 57' 45"\\4-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 293^\circ 57' 45"\)
3. \(6\\ \text{ Dit complex getal ligt op het positief gedeelte van de x-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }6\\\alpha = 0 ^\circ \\\)
4. \(-6+2i\\ r = \sqrt{(-6)^2+2^2} = \sqrt{40} \\ \alpha = tan^{-1}(\frac{2}{-6}) \Leftrightarrow \alpha =161^\circ 33' 54{,}2"\text{ of } \alpha = 341^\circ 33' 54{,}2"\\-6+2i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 161^\circ 33' 54{,}2"\)
5. \(-7-3i\\ r = \sqrt{(-7)^2+(-3)^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{-3}{-7}) \Leftrightarrow \alpha =23^\circ 11' 54{,}9"\text{ of } \alpha = 203^\circ 11' 54{,}9"\\-7-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 203^\circ 11' 54{,}9"\)
6. \(2-i\\ r = \sqrt{2^2+(-1)^2} = \sqrt{5} \\ \alpha = tan^{-1}(\frac{-1}{2}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\2-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 333^\circ 26' 5{,}8"\)
7. \(8-4i\\ r = \sqrt{8^2+(-4)^2} = \sqrt{80} \\ \alpha = tan^{-1}(\frac{-4}{8}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\8-4i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 333^\circ 26' 5{,}8"\)
8. \(-9-5i\\ r = \sqrt{(-9)^2+(-5)^2} = \sqrt{106} \\ \alpha = tan^{-1}(\frac{-5}{-9}) \Leftrightarrow \alpha =29^\circ 3' 16{,}6"\text{ of } \alpha = 209^\circ 3' 16{,}6"\\-9-5i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 209^\circ 3' 16{,}6"\)
9. \(-10+5i\\ r = \sqrt{(-10)^2+5^2} = \sqrt{125} \\ \alpha = tan^{-1}(\frac{5}{-10}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\-10+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 153^\circ 26' 5{,}8"\)
10. \(6+6i\\ r = \sqrt{6^2+6^2} = \sqrt{72} \\ \alpha = tan^{-1}(\frac{6}{6}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\6+6i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
11. \(10-5i\\ r = \sqrt{10^2+(-5)^2} = \sqrt{125} \\ \alpha = tan^{-1}(\frac{-5}{10}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\10-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 333^\circ 26' 5{,}8"\)
12. \(-10+2i\\ r = \sqrt{(-10)^2+2^2} = \sqrt{104} \\ \alpha = tan^{-1}(\frac{2}{-10}) \Leftrightarrow \alpha =168^\circ 41' 24{,}2"\text{ of } \alpha = 348^\circ 41' 24{,}2"\\-10+2i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 168^\circ 41' 24{,}2"\)