Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(5+4i\\ r = \sqrt{5^2+4^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{4}{5}) \Leftrightarrow \alpha =38^\circ 39' 35{,}3"\text{ of } \alpha = 218^\circ 39' 35{,}3"\\5+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 38^\circ 39' 35{,}3"\)
2. \(-10+10i\\ r = \sqrt{(-10)^2+10^2} = \sqrt{200} \\ \alpha = tan^{-1}(\frac{10}{-10}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-10+10i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)
3. \(-2+i\\ r = \sqrt{(-2)^2+1^2} = \sqrt{5} \\ \alpha = tan^{-1}(\frac{1}{-2}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\-2+i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 153^\circ 26' 5{,}8"\)
4. \(4+8i\\ r = \sqrt{4^2+8^2} = \sqrt{80} \\ \alpha = tan^{-1}(\frac{8}{4}) \Leftrightarrow \alpha =63^\circ 26' 5{,}8"\text{ of } \alpha = 243^\circ 26' 5{,}8"\\4+8i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 63^\circ 26' 5{,}8"\)
5. \(4+i\\ r = \sqrt{4^2+1^2} = \sqrt{17} \\ \alpha = tan^{-1}(\frac{1}{4}) \Leftrightarrow \alpha =14^\circ 2' 10{,}5"\text{ of } \alpha = 194^\circ 2' 10{,}5"\\4+i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 14^\circ 2' 10{,}5"\)
6. \(3+4i\\ r = \sqrt{3^2+4^2} = \sqrt{25} \\ \alpha = tan^{-1}(\frac{4}{3}) \Leftrightarrow \alpha =53^\circ 7' 48{,}4"\text{ of } \alpha = 233^\circ 7' 48{,}4"\\3+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 53^\circ 7' 48{,}4"\)
7. \(-1+9i\\ r = \sqrt{(-1)^2+9^2} = \sqrt{82} \\ \alpha = tan^{-1}(\frac{9}{-1}) \Leftrightarrow \alpha =96^\circ 20' 24{,}7"\text{ of } \alpha = 276^\circ 20' 24{,}7"\\-1+9i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 96^\circ 20' 24{,}7"\)
8. \(-9i\\ \text{ Dit complex getal ligt op het negatief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }9\\\alpha = 270 ^\circ \\\)
9. \(-3+9i\\ r = \sqrt{(-3)^2+9^2} = \sqrt{90} \\ \alpha = tan^{-1}(\frac{9}{-3}) \Leftrightarrow \alpha =108^\circ 26' 5{,}8"\text{ of } \alpha = 288^\circ 26' 5{,}8"\\-3+9i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 108^\circ 26' 5{,}8"\)
10. \(-1-6i\\ r = \sqrt{(-1)^2+(-6)^2} = \sqrt{37} \\ \alpha = tan^{-1}(\frac{-6}{-1}) \Leftrightarrow \alpha =80^\circ 32' 15{,}6"\text{ of } \alpha = 260^\circ 32' 15{,}6"\\-1-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 260^\circ 32' 15{,}6"\)
11. \(-4+2i\\ r = \sqrt{(-4)^2+2^2} = \sqrt{20} \\ \alpha = tan^{-1}(\frac{2}{-4}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\-4+2i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 153^\circ 26' 5{,}8"\)
12. \(8+4i\\ r = \sqrt{8^2+4^2} = \sqrt{80} \\ \alpha = tan^{-1}(\frac{4}{8}) \Leftrightarrow \alpha =26^\circ 33' 54{,}2"\text{ of } \alpha = 206^\circ 33' 54{,}2"\\8+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 26^\circ 33' 54{,}2"\)