Bepaal modulus en argument

Hoofdmenu Eentje per keer  🖨

Bepaal modulus en argument

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

---

Bepaal modulus en argument

Verbetersleutel

1. \(3-5i\\ r = \sqrt{3^2+(-5)^2} = \sqrt{34} \\ \alpha = tan^{-1}(\frac{-5}{3}) \Leftrightarrow \alpha =120^\circ 57' 49{,}5"\text{ of } \alpha = 300^\circ 57' 49{,}5"\\3-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 300^\circ 57' 49{,}5"\)
2. \(-7-7i\\ r = \sqrt{(-7)^2+(-7)^2} = \sqrt{98} \\ \alpha = tan^{-1}(\frac{-7}{-7}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\-7-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 225^\circ \)
3. \(8-10i\\ r = \sqrt{8^2+(-10)^2} = \sqrt{164} \\ \alpha = tan^{-1}(\frac{-10}{8}) \Leftrightarrow \alpha =128^\circ 39' 35{,}3"\text{ of } \alpha = 308^\circ 39' 35{,}3"\\8-10i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 308^\circ 39' 35{,}3"\)
4. \(-2+6i\\ r = \sqrt{(-2)^2+6^2} = \sqrt{40} \\ \alpha = tan^{-1}(\frac{6}{-2}) \Leftrightarrow \alpha =108^\circ 26' 5{,}8"\text{ of } \alpha = 288^\circ 26' 5{,}8"\\-2+6i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 108^\circ 26' 5{,}8"\)
5. \(-8+4i\\ r = \sqrt{(-8)^2+4^2} = \sqrt{80} \\ \alpha = tan^{-1}(\frac{4}{-8}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\-8+4i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 153^\circ 26' 5{,}8"\)
6. \(5+6i\\ r = \sqrt{5^2+6^2} = \sqrt{61} \\ \alpha = tan^{-1}(\frac{6}{5}) \Leftrightarrow \alpha =50^\circ 11' 39{,}9"\text{ of } \alpha = 230^\circ 11' 39{,}9"\\5+6i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 50^\circ 11' 39{,}9"\)
7. \(-10+4i\\ r = \sqrt{(-10)^2+4^2} = \sqrt{116} \\ \alpha = tan^{-1}(\frac{4}{-10}) \Leftrightarrow \alpha =158^\circ 11' 54{,}9"\text{ of } \alpha = 338^\circ 11' 54{,}9"\\-10+4i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 158^\circ 11' 54{,}9"\)
8. \(2-8i\\ r = \sqrt{2^2+(-8)^2} = \sqrt{68} \\ \alpha = tan^{-1}(\frac{-8}{2}) \Leftrightarrow \alpha =104^\circ 2' 10{,}5"\text{ of } \alpha = 284^\circ 2' 10{,}5"\\2-8i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 284^\circ 2' 10{,}5"\)
9. \(3-3i\\ r = \sqrt{3^2+(-3)^2} = \sqrt{18} \\ \alpha = tan^{-1}(\frac{-3}{3}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\3-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 315^\circ \)
10. \(10i\\ \text{ Dit complex getal ligt op het positief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }10\\\alpha = 90 ^\circ \\\)
11. \(7-5i\\ r = \sqrt{7^2+(-5)^2} = \sqrt{74} \\ \alpha = tan^{-1}(\frac{-5}{7}) \Leftrightarrow \alpha =144^\circ 27' 44{,}4"\text{ of } \alpha = 324^\circ 27' 44{,}4"\\7-5i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 324^\circ 27' 44{,}4"\)
12. \(7-7i\\ r = \sqrt{7^2+(-7)^2} = \sqrt{98} \\ \alpha = tan^{-1}(\frac{-7}{7}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\7-7i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 315^\circ \)