Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-4-8i\\ r = \sqrt{(-4)^2+(-8)^2} = \sqrt{80} \\ \alpha = tan^{-1}(\frac{-8}{-4}) \Leftrightarrow \alpha =63^\circ 26' 5{,}8"\text{ of } \alpha = 243^\circ 26' 5{,}8"\\-4-8i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 243^\circ 26' 5{,}8"\)
2. \(-5i\\ \text{ Dit complex getal ligt op het negatief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }5\\\alpha = 270 ^\circ \\\)
3. \(-9-i\\ r = \sqrt{(-9)^2+(-1)^2} = \sqrt{82} \\ \alpha = tan^{-1}(\frac{-1}{-9}) \Leftrightarrow \alpha =6^\circ 20' 24{,}7"\text{ of } \alpha = 186^\circ 20' 24{,}7"\\-9-i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 186^\circ 20' 24{,}7"\)
4. \(-9+3i\\ r = \sqrt{(-9)^2+3^2} = \sqrt{90} \\ \alpha = tan^{-1}(\frac{3}{-9}) \Leftrightarrow \alpha =161^\circ 33' 54{,}2"\text{ of } \alpha = 341^\circ 33' 54{,}2"\\-9+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 161^\circ 33' 54{,}2"\)
5. \(-6-2i\\ r = \sqrt{(-6)^2+(-2)^2} = \sqrt{40} \\ \alpha = tan^{-1}(\frac{-2}{-6}) \Leftrightarrow \alpha =18^\circ 26' 5{,}8"\text{ of } \alpha = 198^\circ 26' 5{,}8"\\-6-2i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 198^\circ 26' 5{,}8"\)
6. \(-7+4i\\ r = \sqrt{(-7)^2+4^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{4}{-7}) \Leftrightarrow \alpha =150^\circ 15' 18{,}4"\text{ of } \alpha = 330^\circ 15' 18{,}4"\\-7+4i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 150^\circ 15' 18{,}4"\)
7. \(10+3i\\ r = \sqrt{10^2+3^2} = \sqrt{109} \\ \alpha = tan^{-1}(\frac{3}{10}) \Leftrightarrow \alpha =16^\circ 41' 57{,}3"\text{ of } \alpha = 196^\circ 41' 57{,}3"\\10+3i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 16^\circ 41' 57{,}3"\)
8. \(-4+8i\\ r = \sqrt{(-4)^2+8^2} = \sqrt{80} \\ \alpha = tan^{-1}(\frac{8}{-4}) \Leftrightarrow \alpha =116^\circ 33' 54{,}2"\text{ of } \alpha = 296^\circ 33' 54{,}2"\\-4+8i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 116^\circ 33' 54{,}2"\)
9. \(-10-7i\\ r = \sqrt{(-10)^2+(-7)^2} = \sqrt{149} \\ \alpha = tan^{-1}(\frac{-7}{-10}) \Leftrightarrow \alpha =34^\circ 59' 31{,}3"\text{ of } \alpha = 214^\circ 59' 31{,}3"\\-10-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 214^\circ 59' 31{,}3"\)
10. \(9-7i\\ r = \sqrt{9^2+(-7)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-7}{9}) \Leftrightarrow \alpha =142^\circ 7' 30{,}1"\text{ of } \alpha = 322^\circ 7' 30{,}1"\\9-7i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 322^\circ 7' 30{,}1"\)
11. \(1+5i\\ r = \sqrt{1^2+5^2} = \sqrt{26} \\ \alpha = tan^{-1}(\frac{5}{1}) \Leftrightarrow \alpha =78^\circ 41' 24{,}2"\text{ of } \alpha = 258^\circ 41' 24{,}2"\\1+5i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 78^\circ 41' 24{,}2"\)
12. \(-7-9i\\ r = \sqrt{(-7)^2+(-9)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-9}{-7}) \Leftrightarrow \alpha =52^\circ 7' 30{,}1"\text{ of } \alpha = 232^\circ 7' 30{,}1"\\-7-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 232^\circ 7' 30{,}1"\)