Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-4-10i\\ r = \sqrt{(-4)^2+(-10)^2} = \sqrt{116} \\ \alpha = tan^{-1}(\frac{-10}{-4}) \Leftrightarrow \alpha =68^\circ 11' 54{,}9"\text{ of } \alpha = 248^\circ 11' 54{,}9"\\-4-10i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 248^\circ 11' 54{,}9"\)
2. \(-2+5i\\ r = \sqrt{(-2)^2+5^2} = \sqrt{29} \\ \alpha = tan^{-1}(\frac{5}{-2}) \Leftrightarrow \alpha =111^\circ 48' 5{,}1"\text{ of } \alpha = 291^\circ 48' 5{,}1"\\-2+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 111^\circ 48' 5{,}1"\)
3. \(9-9i\\ r = \sqrt{9^2+(-9)^2} = \sqrt{162} \\ \alpha = tan^{-1}(\frac{-9}{9}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\9-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 315^\circ \)
4. \(-9+4i\\ r = \sqrt{(-9)^2+4^2} = \sqrt{97} \\ \alpha = tan^{-1}(\frac{4}{-9}) \Leftrightarrow \alpha =156^\circ 2' 15"\text{ of } \alpha = 336^\circ 2' 15"\\-9+4i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 156^\circ 2' 15"\)
5. \(5-2i\\ r = \sqrt{5^2+(-2)^2} = \sqrt{29} \\ \alpha = tan^{-1}(\frac{-2}{5}) \Leftrightarrow \alpha =158^\circ 11' 54{,}9"\text{ of } \alpha = 338^\circ 11' 54{,}9"\\5-2i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 338^\circ 11' 54{,}9"\)
6. \(-1+9i\\ r = \sqrt{(-1)^2+9^2} = \sqrt{82} \\ \alpha = tan^{-1}(\frac{9}{-1}) \Leftrightarrow \alpha =96^\circ 20' 24{,}7"\text{ of } \alpha = 276^\circ 20' 24{,}7"\\-1+9i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 96^\circ 20' 24{,}7"\)
7. \(-5-9i\\ r = \sqrt{(-5)^2+(-9)^2} = \sqrt{106} \\ \alpha = tan^{-1}(\frac{-9}{-5}) \Leftrightarrow \alpha =60^\circ 56' 43{,}4"\text{ of } \alpha = 240^\circ 56' 43{,}4"\\-5-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 240^\circ 56' 43{,}4"\)
8. \(-9+9i\\ r = \sqrt{(-9)^2+9^2} = \sqrt{162} \\ \alpha = tan^{-1}(\frac{9}{-9}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-9+9i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)
9. \(7-9i\\ r = \sqrt{7^2+(-9)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-9}{7}) \Leftrightarrow \alpha =127^\circ 52' 29{,}9"\text{ of } \alpha = 307^\circ 52' 29{,}9"\\7-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 307^\circ 52' 29{,}9"\)
10. \(-7-7i\\ r = \sqrt{(-7)^2+(-7)^2} = \sqrt{98} \\ \alpha = tan^{-1}(\frac{-7}{-7}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\-7-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 225^\circ \)
11. \(-4+4i\\ r = \sqrt{(-4)^2+4^2} = \sqrt{32} \\ \alpha = tan^{-1}(\frac{4}{-4}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-4+4i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)
12. \(-8-8i\\ r = \sqrt{(-8)^2+(-8)^2} = \sqrt{128} \\ \alpha = tan^{-1}(\frac{-8}{-8}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\-8-8i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 225^\circ \)