Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-9-7i\\ r = \sqrt{(-9)^2+(-7)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-7}{-9}) \Leftrightarrow \alpha =37^\circ 52' 29{,}9"\text{ of } \alpha = 217^\circ 52' 29{,}9"\\-9-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 217^\circ 52' 29{,}9"\)
2. \(-9+i\\ r = \sqrt{(-9)^2+1^2} = \sqrt{82} \\ \alpha = tan^{-1}(\frac{1}{-9}) \Leftrightarrow \alpha =173^\circ 39' 35{,}3"\text{ of } \alpha = 353^\circ 39' 35{,}3"\\-9+i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 173^\circ 39' 35{,}3"\)
3. \(9+7i\\ r = \sqrt{9^2+7^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{7}{9}) \Leftrightarrow \alpha =37^\circ 52' 29{,}9"\text{ of } \alpha = 217^\circ 52' 29{,}9"\\9+7i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 37^\circ 52' 29{,}9"\)
4. \(-3+10i\\ r = \sqrt{(-3)^2+10^2} = \sqrt{109} \\ \alpha = tan^{-1}(\frac{10}{-3}) \Leftrightarrow \alpha =106^\circ 41' 57{,}3"\text{ of } \alpha = 286^\circ 41' 57{,}3"\\-3+10i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 106^\circ 41' 57{,}3"\)
5. \(8-i\\ r = \sqrt{8^2+(-1)^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{-1}{8}) \Leftrightarrow \alpha =172^\circ 52' 29{,}9"\text{ of } \alpha = 352^\circ 52' 29{,}9"\\8-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 352^\circ 52' 29{,}9"\)
6. \(-3i\\ \text{ Dit complex getal ligt op het negatief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }3\\\alpha = 270 ^\circ \\\)
7. \(9+9i\\ r = \sqrt{9^2+9^2} = \sqrt{162} \\ \alpha = tan^{-1}(\frac{9}{9}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\9+9i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
8. \(-2+5i\\ r = \sqrt{(-2)^2+5^2} = \sqrt{29} \\ \alpha = tan^{-1}(\frac{5}{-2}) \Leftrightarrow \alpha =111^\circ 48' 5{,}1"\text{ of } \alpha = 291^\circ 48' 5{,}1"\\-2+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 111^\circ 48' 5{,}1"\)
9. \(7-i\\ r = \sqrt{7^2+(-1)^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{-1}{7}) \Leftrightarrow \alpha =171^\circ 52' 11{,}6"\text{ of } \alpha = 351^\circ 52' 11{,}6"\\7-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 351^\circ 52' 11{,}6"\)
10. \(-7-8i\\ r = \sqrt{(-7)^2+(-8)^2} = \sqrt{113} \\ \alpha = tan^{-1}(\frac{-8}{-7}) \Leftrightarrow \alpha =48^\circ 48' 50{,}7"\text{ of } \alpha = 228^\circ 48' 50{,}7"\\-7-8i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 228^\circ 48' 50{,}7"\)
11. \(1+i\\ r = \sqrt{1^2+1^2} = \sqrt{2} \\ \alpha = tan^{-1}(\frac{1}{1}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\1+i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
12. \(10-8i\\ r = \sqrt{10^2+(-8)^2} = \sqrt{164} \\ \alpha = tan^{-1}(\frac{-8}{10}) \Leftrightarrow \alpha =141^\circ 20' 24{,}7"\text{ of } \alpha = 321^\circ 20' 24{,}7"\\10-8i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 321^\circ 20' 24{,}7"\)