Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(5-4i\\ r = \sqrt{5^2+(-4)^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{-4}{5}) \Leftrightarrow \alpha =141^\circ 20' 24{,}7"\text{ of } \alpha = 321^\circ 20' 24{,}7"\\5-4i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 321^\circ 20' 24{,}7"\)
2. \(-5-4i\\ r = \sqrt{(-5)^2+(-4)^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{-4}{-5}) \Leftrightarrow \alpha =38^\circ 39' 35{,}3"\text{ of } \alpha = 218^\circ 39' 35{,}3"\\-5-4i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 218^\circ 39' 35{,}3"\)
3. \(-6-2i\\ r = \sqrt{(-6)^2+(-2)^2} = \sqrt{40} \\ \alpha = tan^{-1}(\frac{-2}{-6}) \Leftrightarrow \alpha =18^\circ 26' 5{,}8"\text{ of } \alpha = 198^\circ 26' 5{,}8"\\-6-2i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 198^\circ 26' 5{,}8"\)
4. \(-10+3i\\ r = \sqrt{(-10)^2+3^2} = \sqrt{109} \\ \alpha = tan^{-1}(\frac{3}{-10}) \Leftrightarrow \alpha =163^\circ 18' 2{,}7"\text{ of } \alpha = 343^\circ 18' 2{,}7"\\-10+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 163^\circ 18' 2{,}7"\)
5. \(7\\ \text{ Dit complex getal ligt op het positief gedeelte van de x-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }7\\\alpha = 0 ^\circ \\\)
6. \(2-3i\\ r = \sqrt{2^2+(-3)^2} = \sqrt{13} \\ \alpha = tan^{-1}(\frac{-3}{2}) \Leftrightarrow \alpha =123^\circ 41' 24{,}2"\text{ of } \alpha = 303^\circ 41' 24{,}2"\\2-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 303^\circ 41' 24{,}2"\)
7. \(3+4i\\ r = \sqrt{3^2+4^2} = \sqrt{25} \\ \alpha = tan^{-1}(\frac{4}{3}) \Leftrightarrow \alpha =53^\circ 7' 48{,}4"\text{ of } \alpha = 233^\circ 7' 48{,}4"\\3+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 53^\circ 7' 48{,}4"\)
8. \(-2+7i\\ r = \sqrt{(-2)^2+7^2} = \sqrt{53} \\ \alpha = tan^{-1}(\frac{7}{-2}) \Leftrightarrow \alpha =105^\circ 56' 43{,}4"\text{ of } \alpha = 285^\circ 56' 43{,}4"\\-2+7i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 105^\circ 56' 43{,}4"\)
9. \(5+i\\ r = \sqrt{5^2+1^2} = \sqrt{26} \\ \alpha = tan^{-1}(\frac{1}{5}) \Leftrightarrow \alpha =11^\circ 18' 35{,}8"\text{ of } \alpha = 191^\circ 18' 35{,}8"\\5+i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 11^\circ 18' 35{,}8"\)
10. \(-2-3i\\ r = \sqrt{(-2)^2+(-3)^2} = \sqrt{13} \\ \alpha = tan^{-1}(\frac{-3}{-2}) \Leftrightarrow \alpha =56^\circ 18' 35{,}8"\text{ of } \alpha = 236^\circ 18' 35{,}8"\\-2-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 236^\circ 18' 35{,}8"\)
11. \(5-5i\\ r = \sqrt{5^2+(-5)^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{-5}{5}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\5-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 315^\circ \)
12. \(-4+i\\ r = \sqrt{(-4)^2+1^2} = \sqrt{17} \\ \alpha = tan^{-1}(\frac{1}{-4}) \Leftrightarrow \alpha =165^\circ 57' 49{,}5"\text{ of } \alpha = 345^\circ 57' 49{,}5"\\-4+i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 165^\circ 57' 49{,}5"\)