Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-6+9i\\ r = \sqrt{(-6)^2+9^2} = \sqrt{117} \\ \alpha = tan^{-1}(\frac{9}{-6}) \Leftrightarrow \alpha =123^\circ 41' 24{,}2"\text{ of } \alpha = 303^\circ 41' 24{,}2"\\-6+9i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 123^\circ 41' 24{,}2"\)
2. \(3+9i\\ r = \sqrt{3^2+9^2} = \sqrt{90} \\ \alpha = tan^{-1}(\frac{9}{3}) \Leftrightarrow \alpha =71^\circ 33' 54{,}2"\text{ of } \alpha = 251^\circ 33' 54{,}2"\\3+9i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 71^\circ 33' 54{,}2"\)
3. \(-4+3i\\ r = \sqrt{(-4)^2+3^2} = \sqrt{25} \\ \alpha = tan^{-1}(\frac{3}{-4}) \Leftrightarrow \alpha =143^\circ 7' 48{,}4"\text{ of } \alpha = 323^\circ 7' 48{,}4"\\-4+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 143^\circ 7' 48{,}4"\)
4. \(1-3i\\ r = \sqrt{1^2+(-3)^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{-3}{1}) \Leftrightarrow \alpha =108^\circ 26' 5{,}8"\text{ of } \alpha = 288^\circ 26' 5{,}8"\\1-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 288^\circ 26' 5{,}8"\)
5. \(-8-6i\\ r = \sqrt{(-8)^2+(-6)^2} = \sqrt{100} \\ \alpha = tan^{-1}(\frac{-6}{-8}) \Leftrightarrow \alpha =36^\circ 52' 11{,}6"\text{ of } \alpha = 216^\circ 52' 11{,}6"\\-8-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 216^\circ 52' 11{,}6"\)
6. \(4+7i\\ r = \sqrt{4^2+7^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{7}{4}) \Leftrightarrow \alpha =60^\circ 15' 18{,}4"\text{ of } \alpha = 240^\circ 15' 18{,}4"\\4+7i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 60^\circ 15' 18{,}4"\)
7. \(4-9i\\ r = \sqrt{4^2+(-9)^2} = \sqrt{97} \\ \alpha = tan^{-1}(\frac{-9}{4}) \Leftrightarrow \alpha =113^\circ 57' 45"\text{ of } \alpha = 293^\circ 57' 45"\\4-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 293^\circ 57' 45"\)
8. \(-9-i\\ r = \sqrt{(-9)^2+(-1)^2} = \sqrt{82} \\ \alpha = tan^{-1}(\frac{-1}{-9}) \Leftrightarrow \alpha =6^\circ 20' 24{,}7"\text{ of } \alpha = 186^\circ 20' 24{,}7"\\-9-i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 186^\circ 20' 24{,}7"\)
9. \(-10-5i\\ r = \sqrt{(-10)^2+(-5)^2} = \sqrt{125} \\ \alpha = tan^{-1}(\frac{-5}{-10}) \Leftrightarrow \alpha =26^\circ 33' 54{,}2"\text{ of } \alpha = 206^\circ 33' 54{,}2"\\-10-5i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 206^\circ 33' 54{,}2"\)
10. \(7-i\\ r = \sqrt{7^2+(-1)^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{-1}{7}) \Leftrightarrow \alpha =171^\circ 52' 11{,}6"\text{ of } \alpha = 351^\circ 52' 11{,}6"\\7-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 351^\circ 52' 11{,}6"\)
11. \(-7-9i\\ r = \sqrt{(-7)^2+(-9)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-9}{-7}) \Leftrightarrow \alpha =52^\circ 7' 30{,}1"\text{ of } \alpha = 232^\circ 7' 30{,}1"\\-7-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 232^\circ 7' 30{,}1"\)
12. \(9-9i\\ r = \sqrt{9^2+(-9)^2} = \sqrt{162} \\ \alpha = tan^{-1}(\frac{-9}{9}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\9-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 315^\circ \)