Bepaal modulus en argument

Hoofdmenu Eentje per keer  🖨

Bepaal modulus en argument

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

---

Bepaal modulus en argument

Verbetersleutel

1. \(9+10i\\ r = \sqrt{9^2+10^2} = \sqrt{181} \\ \alpha = tan^{-1}(\frac{10}{9}) \Leftrightarrow \alpha =48^\circ 0' 46"\text{ of } \alpha = 228^\circ 0' 46"\\9+10i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 48^\circ 0' 46"\)
2. \(-4-8i\\ r = \sqrt{(-4)^2+(-8)^2} = \sqrt{80} \\ \alpha = tan^{-1}(\frac{-8}{-4}) \Leftrightarrow \alpha =63^\circ 26' 5{,}8"\text{ of } \alpha = 243^\circ 26' 5{,}8"\\-4-8i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 243^\circ 26' 5{,}8"\)
3. \(-10+10i\\ r = \sqrt{(-10)^2+10^2} = \sqrt{200} \\ \alpha = tan^{-1}(\frac{10}{-10}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-10+10i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)
4. \(-1+3i\\ r = \sqrt{(-1)^2+3^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{3}{-1}) \Leftrightarrow \alpha =108^\circ 26' 5{,}8"\text{ of } \alpha = 288^\circ 26' 5{,}8"\\-1+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 108^\circ 26' 5{,}8"\)
5. \(-9-10i\\ r = \sqrt{(-9)^2+(-10)^2} = \sqrt{181} \\ \alpha = tan^{-1}(\frac{-10}{-9}) \Leftrightarrow \alpha =48^\circ 0' 46"\text{ of } \alpha = 228^\circ 0' 46"\\-9-10i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 228^\circ 0' 46"\)
6. \(2-i\\ r = \sqrt{2^2+(-1)^2} = \sqrt{5} \\ \alpha = tan^{-1}(\frac{-1}{2}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\2-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 333^\circ 26' 5{,}8"\)
7. \(-2-6i\\ r = \sqrt{(-2)^2+(-6)^2} = \sqrt{40} \\ \alpha = tan^{-1}(\frac{-6}{-2}) \Leftrightarrow \alpha =71^\circ 33' 54{,}2"\text{ of } \alpha = 251^\circ 33' 54{,}2"\\-2-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 251^\circ 33' 54{,}2"\)
8. \(8+7i\\ r = \sqrt{8^2+7^2} = \sqrt{113} \\ \alpha = tan^{-1}(\frac{7}{8}) \Leftrightarrow \alpha =41^\circ 11' 9{,}3"\text{ of } \alpha = 221^\circ 11' 9{,}3"\\8+7i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 41^\circ 11' 9{,}3"\)
9. \(-10\\ \text{ Dit complex getal ligt op het negatief gedeelte van de x-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }10\\\alpha = 180 ^\circ \\\)
10. \(-1+3i\\ r = \sqrt{(-1)^2+3^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{3}{-1}) \Leftrightarrow \alpha =108^\circ 26' 5{,}8"\text{ of } \alpha = 288^\circ 26' 5{,}8"\\-1+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 108^\circ 26' 5{,}8"\)
11. \(-4-4i\\ r = \sqrt{(-4)^2+(-4)^2} = \sqrt{32} \\ \alpha = tan^{-1}(\frac{-4}{-4}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\-4-4i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 225^\circ \)
12. \(9+i\\ r = \sqrt{9^2+1^2} = \sqrt{82} \\ \alpha = tan^{-1}(\frac{1}{9}) \Leftrightarrow \alpha =6^\circ 20' 24{,}7"\text{ of } \alpha = 186^\circ 20' 24{,}7"\\9+i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 6^\circ 20' 24{,}7"\)