Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-7-10i\\ r = \sqrt{(-7)^2+(-10)^2} = \sqrt{149} \\ \alpha = tan^{-1}(\frac{-10}{-7}) \Leftrightarrow \alpha =55^\circ 0' 28{,}7"\text{ of } \alpha = 235^\circ 0' 28{,}7"\\-7-10i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 235^\circ 0' 28{,}7"\)
2. \(-5-5i\\ r = \sqrt{(-5)^2+(-5)^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{-5}{-5}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\-5-5i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 225^\circ \)
3. \(5-5i\\ r = \sqrt{5^2+(-5)^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{-5}{5}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\5-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 315^\circ \)
4. \(3-7i\\ r = \sqrt{3^2+(-7)^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{-7}{3}) \Leftrightarrow \alpha =113^\circ 11' 54{,}9"\text{ of } \alpha = 293^\circ 11' 54{,}9"\\3-7i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 293^\circ 11' 54{,}9"\)
5. \(-6-3i\\ r = \sqrt{(-6)^2+(-3)^2} = \sqrt{45} \\ \alpha = tan^{-1}(\frac{-3}{-6}) \Leftrightarrow \alpha =26^\circ 33' 54{,}2"\text{ of } \alpha = 206^\circ 33' 54{,}2"\\-6-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 206^\circ 33' 54{,}2"\)
6. \(7-5i\\ r = \sqrt{7^2+(-5)^2} = \sqrt{74} \\ \alpha = tan^{-1}(\frac{-5}{7}) \Leftrightarrow \alpha =144^\circ 27' 44{,}4"\text{ of } \alpha = 324^\circ 27' 44{,}4"\\7-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 324^\circ 27' 44{,}4"\)
7. \(-4-i\\ r = \sqrt{(-4)^2+(-1)^2} = \sqrt{17} \\ \alpha = tan^{-1}(\frac{-1}{-4}) \Leftrightarrow \alpha =14^\circ 2' 10{,}5"\text{ of } \alpha = 194^\circ 2' 10{,}5"\\-4-i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 194^\circ 2' 10{,}5"\)
8. \(8+3i\\ r = \sqrt{8^2+3^2} = \sqrt{73} \\ \alpha = tan^{-1}(\frac{3}{8}) \Leftrightarrow \alpha =20^\circ 33' 21{,}8"\text{ of } \alpha = 200^\circ 33' 21{,}8"\\8+3i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 20^\circ 33' 21{,}8"\)
9. \(6-5i\\ r = \sqrt{6^2+(-5)^2} = \sqrt{61} \\ \alpha = tan^{-1}(\frac{-5}{6}) \Leftrightarrow \alpha =140^\circ 11' 39{,}9"\text{ of } \alpha = 320^\circ 11' 39{,}9"\\6-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 320^\circ 11' 39{,}9"\)
10. \(-5i\\ \text{ Dit complex getal ligt op het negatief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }5\\\alpha = 270 ^\circ \\\)
11. \(9-10i\\ r = \sqrt{9^2+(-10)^2} = \sqrt{181} \\ \alpha = tan^{-1}(\frac{-10}{9}) \Leftrightarrow \alpha =131^\circ 59' 14"\text{ of } \alpha = 311^\circ 59' 14"\\9-10i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 311^\circ 59' 14"\)
12. \(4+4i\\ r = \sqrt{4^2+4^2} = \sqrt{32} \\ \alpha = tan^{-1}(\frac{4}{4}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\4+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)