Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-9-6i\\ r = \sqrt{(-9)^2+(-6)^2} = \sqrt{117} \\ \alpha = tan^{-1}(\frac{-6}{-9}) \Leftrightarrow \alpha =33^\circ 41' 24{,}2"\text{ of } \alpha = 213^\circ 41' 24{,}2"\\-9-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 213^\circ 41' 24{,}2"\)
2. \(-3i\\ \text{ Dit complex getal ligt op het negatief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }3\\\alpha = 270 ^\circ \\\)
3. \(-9+7i\\ r = \sqrt{(-9)^2+7^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{7}{-9}) \Leftrightarrow \alpha =142^\circ 7' 30{,}1"\text{ of } \alpha = 322^\circ 7' 30{,}1"\\-9+7i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 142^\circ 7' 30{,}1"\)
4. \(6-8i\\ r = \sqrt{6^2+(-8)^2} = \sqrt{100} \\ \alpha = tan^{-1}(\frac{-8}{6}) \Leftrightarrow \alpha =126^\circ 52' 11{,}6"\text{ of } \alpha = 306^\circ 52' 11{,}6"\\6-8i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 306^\circ 52' 11{,}6"\)
5. \(3+9i\\ r = \sqrt{3^2+9^2} = \sqrt{90} \\ \alpha = tan^{-1}(\frac{9}{3}) \Leftrightarrow \alpha =71^\circ 33' 54{,}2"\text{ of } \alpha = 251^\circ 33' 54{,}2"\\3+9i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 71^\circ 33' 54{,}2"\)
6. \(-2-2i\\ r = \sqrt{(-2)^2+(-2)^2} = \sqrt{8} \\ \alpha = tan^{-1}(\frac{-2}{-2}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\-2-2i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 225^\circ \)
7. \(-4-3i\\ r = \sqrt{(-4)^2+(-3)^2} = \sqrt{25} \\ \alpha = tan^{-1}(\frac{-3}{-4}) \Leftrightarrow \alpha =36^\circ 52' 11{,}6"\text{ of } \alpha = 216^\circ 52' 11{,}6"\\-4-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 216^\circ 52' 11{,}6"\)
8. \(-1+9i\\ r = \sqrt{(-1)^2+9^2} = \sqrt{82} \\ \alpha = tan^{-1}(\frac{9}{-1}) \Leftrightarrow \alpha =96^\circ 20' 24{,}7"\text{ of } \alpha = 276^\circ 20' 24{,}7"\\-1+9i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 96^\circ 20' 24{,}7"\)
9. \(-7+4i\\ r = \sqrt{(-7)^2+4^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{4}{-7}) \Leftrightarrow \alpha =150^\circ 15' 18{,}4"\text{ of } \alpha = 330^\circ 15' 18{,}4"\\-7+4i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 150^\circ 15' 18{,}4"\)
10. \(5+4i\\ r = \sqrt{5^2+4^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{4}{5}) \Leftrightarrow \alpha =38^\circ 39' 35{,}3"\text{ of } \alpha = 218^\circ 39' 35{,}3"\\5+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 38^\circ 39' 35{,}3"\)
11. \(-9+2i\\ r = \sqrt{(-9)^2+2^2} = \sqrt{85} \\ \alpha = tan^{-1}(\frac{2}{-9}) \Leftrightarrow \alpha =167^\circ 28' 16{,}3"\text{ of } \alpha = 347^\circ 28' 16{,}3"\\-9+2i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 167^\circ 28' 16{,}3"\)
12. \(-3+10i\\ r = \sqrt{(-3)^2+10^2} = \sqrt{109} \\ \alpha = tan^{-1}(\frac{10}{-3}) \Leftrightarrow \alpha =106^\circ 41' 57{,}3"\text{ of } \alpha = 286^\circ 41' 57{,}3"\\-3+10i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 106^\circ 41' 57{,}3"\)