Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(8-4i\\ r = \sqrt{8^2+(-4)^2} = \sqrt{80} \\ \alpha = tan^{-1}(\frac{-4}{8}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\8-4i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 333^\circ 26' 5{,}8"\)
2. \(10+10i\\ r = \sqrt{10^2+10^2} = \sqrt{200} \\ \alpha = tan^{-1}(\frac{10}{10}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\10+10i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
3. \(-6-4i\\ r = \sqrt{(-6)^2+(-4)^2} = \sqrt{52} \\ \alpha = tan^{-1}(\frac{-4}{-6}) \Leftrightarrow \alpha =33^\circ 41' 24{,}2"\text{ of } \alpha = 213^\circ 41' 24{,}2"\\-6-4i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 213^\circ 41' 24{,}2"\)
4. \(9+3i\\ r = \sqrt{9^2+3^2} = \sqrt{90} \\ \alpha = tan^{-1}(\frac{3}{9}) \Leftrightarrow \alpha =18^\circ 26' 5{,}8"\text{ of } \alpha = 198^\circ 26' 5{,}8"\\9+3i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 18^\circ 26' 5{,}8"\)
5. \(6-10i\\ r = \sqrt{6^2+(-10)^2} = \sqrt{136} \\ \alpha = tan^{-1}(\frac{-10}{6}) \Leftrightarrow \alpha =120^\circ 57' 49{,}5"\text{ of } \alpha = 300^\circ 57' 49{,}5"\\6-10i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 300^\circ 57' 49{,}5"\)
6. \(-5-7i\\ r = \sqrt{(-5)^2+(-7)^2} = \sqrt{74} \\ \alpha = tan^{-1}(\frac{-7}{-5}) \Leftrightarrow \alpha =54^\circ 27' 44{,}4"\text{ of } \alpha = 234^\circ 27' 44{,}4"\\-5-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 234^\circ 27' 44{,}4"\)
7. \(-i\\ \text{ Dit complex getal ligt op het negatief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }1\\\alpha = 270 ^\circ \\\)
8. \(2+2i\\ r = \sqrt{2^2+2^2} = \sqrt{8} \\ \alpha = tan^{-1}(\frac{2}{2}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\2+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
9. \(-2+6i\\ r = \sqrt{(-2)^2+6^2} = \sqrt{40} \\ \alpha = tan^{-1}(\frac{6}{-2}) \Leftrightarrow \alpha =108^\circ 26' 5{,}8"\text{ of } \alpha = 288^\circ 26' 5{,}8"\\-2+6i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 108^\circ 26' 5{,}8"\)
10. \(7-3i\\ r = \sqrt{7^2+(-3)^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{-3}{7}) \Leftrightarrow \alpha =156^\circ 48' 5{,}1"\text{ of } \alpha = 336^\circ 48' 5{,}1"\\7-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 336^\circ 48' 5{,}1"\)
11. \(7-6i\\ r = \sqrt{7^2+(-6)^2} = \sqrt{85} \\ \alpha = tan^{-1}(\frac{-6}{7}) \Leftrightarrow \alpha =139^\circ 23' 55{,}3"\text{ of } \alpha = 319^\circ 23' 55{,}3"\\7-6i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 319^\circ 23' 55{,}3"\)
12. \(-3+4i\\ r = \sqrt{(-3)^2+4^2} = \sqrt{25} \\ \alpha = tan^{-1}(\frac{4}{-3}) \Leftrightarrow \alpha =126^\circ 52' 11{,}6"\text{ of } \alpha = 306^\circ 52' 11{,}6"\\-3+4i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 126^\circ 52' 11{,}6"\)