Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(7i\\ \text{ Dit complex getal ligt op het positief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }7\\\alpha = 90 ^\circ \\\)
2. \(-7+i\\ r = \sqrt{(-7)^2+1^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{1}{-7}) \Leftrightarrow \alpha =171^\circ 52' 11{,}6"\text{ of } \alpha = 351^\circ 52' 11{,}6"\\-7+i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 171^\circ 52' 11{,}6"\)
3. \(7+4i\\ r = \sqrt{7^2+4^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{4}{7}) \Leftrightarrow \alpha =29^\circ 44' 41{,}6"\text{ of } \alpha = 209^\circ 44' 41{,}6"\\7+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 29^\circ 44' 41{,}6"\)
4. \(3-2i\\ r = \sqrt{3^2+(-2)^2} = \sqrt{13} \\ \alpha = tan^{-1}(\frac{-2}{3}) \Leftrightarrow \alpha =146^\circ 18' 35{,}8"\text{ of } \alpha = 326^\circ 18' 35{,}8"\\3-2i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 326^\circ 18' 35{,}8"\)
5. \(1-5i\\ r = \sqrt{1^2+(-5)^2} = \sqrt{26} \\ \alpha = tan^{-1}(\frac{-5}{1}) \Leftrightarrow \alpha =101^\circ 18' 35{,}8"\text{ of } \alpha = 281^\circ 18' 35{,}8"\\1-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 281^\circ 18' 35{,}8"\)
6. \(-4-7i\\ r = \sqrt{(-4)^2+(-7)^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{-7}{-4}) \Leftrightarrow \alpha =60^\circ 15' 18{,}4"\text{ of } \alpha = 240^\circ 15' 18{,}4"\\-4-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 240^\circ 15' 18{,}4"\)
7. \(2+3i\\ r = \sqrt{2^2+3^2} = \sqrt{13} \\ \alpha = tan^{-1}(\frac{3}{2}) \Leftrightarrow \alpha =56^\circ 18' 35{,}8"\text{ of } \alpha = 236^\circ 18' 35{,}8"\\2+3i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 56^\circ 18' 35{,}8"\)
8. \(-7+4i\\ r = \sqrt{(-7)^2+4^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{4}{-7}) \Leftrightarrow \alpha =150^\circ 15' 18{,}4"\text{ of } \alpha = 330^\circ 15' 18{,}4"\\-7+4i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 150^\circ 15' 18{,}4"\)
9. \(2+7i\\ r = \sqrt{2^2+7^2} = \sqrt{53} \\ \alpha = tan^{-1}(\frac{7}{2}) \Leftrightarrow \alpha =74^\circ 3' 16{,}6"\text{ of } \alpha = 254^\circ 3' 16{,}6"\\2+7i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 74^\circ 3' 16{,}6"\)
10. \(-5-6i\\ r = \sqrt{(-5)^2+(-6)^2} = \sqrt{61} \\ \alpha = tan^{-1}(\frac{-6}{-5}) \Leftrightarrow \alpha =50^\circ 11' 39{,}9"\text{ of } \alpha = 230^\circ 11' 39{,}9"\\-5-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 230^\circ 11' 39{,}9"\)
11. \(-10-5i\\ r = \sqrt{(-10)^2+(-5)^2} = \sqrt{125} \\ \alpha = tan^{-1}(\frac{-5}{-10}) \Leftrightarrow \alpha =26^\circ 33' 54{,}2"\text{ of } \alpha = 206^\circ 33' 54{,}2"\\-10-5i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 206^\circ 33' 54{,}2"\)
12. \(-9-7i\\ r = \sqrt{(-9)^2+(-7)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-7}{-9}) \Leftrightarrow \alpha =37^\circ 52' 29{,}9"\text{ of } \alpha = 217^\circ 52' 29{,}9"\\-9-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 217^\circ 52' 29{,}9"\)