Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-4+7i\\ r = \sqrt{(-4)^2+7^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{7}{-4}) \Leftrightarrow \alpha =119^\circ 44' 41{,}6"\text{ of } \alpha = 299^\circ 44' 41{,}6"\\-4+7i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 119^\circ 44' 41{,}6"\)
2. \(10+2i\\ r = \sqrt{10^2+2^2} = \sqrt{104} \\ \alpha = tan^{-1}(\frac{2}{10}) \Leftrightarrow \alpha =11^\circ 18' 35{,}8"\text{ of } \alpha = 191^\circ 18' 35{,}8"\\10+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 11^\circ 18' 35{,}8"\)
3. \(9-2i\\ r = \sqrt{9^2+(-2)^2} = \sqrt{85} \\ \alpha = tan^{-1}(\frac{-2}{9}) \Leftrightarrow \alpha =167^\circ 28' 16{,}3"\text{ of } \alpha = 347^\circ 28' 16{,}3"\\9-2i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 347^\circ 28' 16{,}3"\)
4. \(4+4i\\ r = \sqrt{4^2+4^2} = \sqrt{32} \\ \alpha = tan^{-1}(\frac{4}{4}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\4+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
5. \(-7+7i\\ r = \sqrt{(-7)^2+7^2} = \sqrt{98} \\ \alpha = tan^{-1}(\frac{7}{-7}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-7+7i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)
6. \(-7-8i\\ r = \sqrt{(-7)^2+(-8)^2} = \sqrt{113} \\ \alpha = tan^{-1}(\frac{-8}{-7}) \Leftrightarrow \alpha =48^\circ 48' 50{,}7"\text{ of } \alpha = 228^\circ 48' 50{,}7"\\-7-8i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 228^\circ 48' 50{,}7"\)
7. \(1+i\\ r = \sqrt{1^2+1^2} = \sqrt{2} \\ \alpha = tan^{-1}(\frac{1}{1}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\1+i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
8. \(9-9i\\ r = \sqrt{9^2+(-9)^2} = \sqrt{162} \\ \alpha = tan^{-1}(\frac{-9}{9}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\9-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 315^\circ \)
9. \(10-i\\ r = \sqrt{10^2+(-1)^2} = \sqrt{101} \\ \alpha = tan^{-1}(\frac{-1}{10}) \Leftrightarrow \alpha =174^\circ 17' 21{,}9"\text{ of } \alpha = 354^\circ 17' 21{,}9"\\10-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 354^\circ 17' 21{,}9"\)
10. \(9+9i\\ r = \sqrt{9^2+9^2} = \sqrt{162} \\ \alpha = tan^{-1}(\frac{9}{9}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\9+9i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
11. \(-1-3i\\ r = \sqrt{(-1)^2+(-3)^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{-3}{-1}) \Leftrightarrow \alpha =71^\circ 33' 54{,}2"\text{ of } \alpha = 251^\circ 33' 54{,}2"\\-1-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 251^\circ 33' 54{,}2"\)
12. \(-4+2i\\ r = \sqrt{(-4)^2+2^2} = \sqrt{20} \\ \alpha = tan^{-1}(\frac{2}{-4}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\-4+2i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 153^\circ 26' 5{,}8"\)