Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-6-9i\\ r = \sqrt{(-6)^2+(-9)^2} = \sqrt{117} \\ \alpha = tan^{-1}(\frac{-9}{-6}) \Leftrightarrow \alpha =56^\circ 18' 35{,}8"\text{ of } \alpha = 236^\circ 18' 35{,}8"\\-6-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 236^\circ 18' 35{,}8"\)
2. \(5+5i\\ r = \sqrt{5^2+5^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{5}{5}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\5+5i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
3. \(7-3i\\ r = \sqrt{7^2+(-3)^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{-3}{7}) \Leftrightarrow \alpha =156^\circ 48' 5{,}1"\text{ of } \alpha = 336^\circ 48' 5{,}1"\\7-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 336^\circ 48' 5{,}1"\)
4. \(4-7i\\ r = \sqrt{4^2+(-7)^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{-7}{4}) \Leftrightarrow \alpha =119^\circ 44' 41{,}6"\text{ of } \alpha = 299^\circ 44' 41{,}6"\\4-7i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 299^\circ 44' 41{,}6"\)
5. \(-10\\ \text{ Dit complex getal ligt op het negatief gedeelte van de x-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }10\\\alpha = 180 ^\circ \\\)
6. \(-6-i\\ r = \sqrt{(-6)^2+(-1)^2} = \sqrt{37} \\ \alpha = tan^{-1}(\frac{-1}{-6}) \Leftrightarrow \alpha =9^\circ 27' 44{,}4"\text{ of } \alpha = 189^\circ 27' 44{,}4"\\-6-i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 189^\circ 27' 44{,}4"\)
7. \(2+10i\\ r = \sqrt{2^2+10^2} = \sqrt{104} \\ \alpha = tan^{-1}(\frac{10}{2}) \Leftrightarrow \alpha =78^\circ 41' 24{,}2"\text{ of } \alpha = 258^\circ 41' 24{,}2"\\2+10i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 78^\circ 41' 24{,}2"\)
8. \(5-3i\\ r = \sqrt{5^2+(-3)^2} = \sqrt{34} \\ \alpha = tan^{-1}(\frac{-3}{5}) \Leftrightarrow \alpha =149^\circ 2' 10{,}5"\text{ of } \alpha = 329^\circ 2' 10{,}5"\\5-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 329^\circ 2' 10{,}5"\)
9. \(6+2i\\ r = \sqrt{6^2+2^2} = \sqrt{40} \\ \alpha = tan^{-1}(\frac{2}{6}) \Leftrightarrow \alpha =18^\circ 26' 5{,}8"\text{ of } \alpha = 198^\circ 26' 5{,}8"\\6+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 18^\circ 26' 5{,}8"\)
10. \(1+8i\\ r = \sqrt{1^2+8^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{8}{1}) \Leftrightarrow \alpha =82^\circ 52' 29{,}9"\text{ of } \alpha = 262^\circ 52' 29{,}9"\\1+8i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 82^\circ 52' 29{,}9"\)
11. \(-3-i\\ r = \sqrt{(-3)^2+(-1)^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{-1}{-3}) \Leftrightarrow \alpha =18^\circ 26' 5{,}8"\text{ of } \alpha = 198^\circ 26' 5{,}8"\\-3-i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 198^\circ 26' 5{,}8"\)
12. \(-1+7i\\ r = \sqrt{(-1)^2+7^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{7}{-1}) \Leftrightarrow \alpha =98^\circ 7' 48{,}4"\text{ of } \alpha = 278^\circ 7' 48{,}4"\\-1+7i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 98^\circ 7' 48{,}4"\)