Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(7+10i\\ r = \sqrt{7^2+10^2} = \sqrt{149} \\ \alpha = tan^{-1}(\frac{10}{7}) \Leftrightarrow \alpha =55^\circ 0' 28{,}7"\text{ of } \alpha = 235^\circ 0' 28{,}7"\\7+10i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 55^\circ 0' 28{,}7"\)
2. \(-4-3i\\ r = \sqrt{(-4)^2+(-3)^2} = \sqrt{25} \\ \alpha = tan^{-1}(\frac{-3}{-4}) \Leftrightarrow \alpha =36^\circ 52' 11{,}6"\text{ of } \alpha = 216^\circ 52' 11{,}6"\\-4-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 216^\circ 52' 11{,}6"\)
3. \(7-8i\\ r = \sqrt{7^2+(-8)^2} = \sqrt{113} \\ \alpha = tan^{-1}(\frac{-8}{7}) \Leftrightarrow \alpha =131^\circ 11' 9{,}3"\text{ of } \alpha = 311^\circ 11' 9{,}3"\\7-8i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 311^\circ 11' 9{,}3"\)
4. \(8+5i\\ r = \sqrt{8^2+5^2} = \sqrt{89} \\ \alpha = tan^{-1}(\frac{5}{8}) \Leftrightarrow \alpha =32^\circ 0' 19{,}4"\text{ of } \alpha = 212^\circ 0' 19{,}4"\\8+5i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 32^\circ 0' 19{,}4"\)
5. \(-9+8i\\ r = \sqrt{(-9)^2+8^2} = \sqrt{145} \\ \alpha = tan^{-1}(\frac{8}{-9}) \Leftrightarrow \alpha =138^\circ 21' 59{,}3"\text{ of } \alpha = 318^\circ 21' 59{,}3"\\-9+8i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 138^\circ 21' 59{,}3"\)
6. \(7-4i\\ r = \sqrt{7^2+(-4)^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{-4}{7}) \Leftrightarrow \alpha =150^\circ 15' 18{,}4"\text{ of } \alpha = 330^\circ 15' 18{,}4"\\7-4i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 330^\circ 15' 18{,}4"\)
7. \(-5\\ \text{ Dit complex getal ligt op het negatief gedeelte van de x-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }5\\\alpha = 180 ^\circ \\\)
8. \(-3+2i\\ r = \sqrt{(-3)^2+2^2} = \sqrt{13} \\ \alpha = tan^{-1}(\frac{2}{-3}) \Leftrightarrow \alpha =146^\circ 18' 35{,}8"\text{ of } \alpha = 326^\circ 18' 35{,}8"\\-3+2i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 146^\circ 18' 35{,}8"\)
9. \(2+2i\\ r = \sqrt{2^2+2^2} = \sqrt{8} \\ \alpha = tan^{-1}(\frac{2}{2}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\2+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
10. \(5-3i\\ r = \sqrt{5^2+(-3)^2} = \sqrt{34} \\ \alpha = tan^{-1}(\frac{-3}{5}) \Leftrightarrow \alpha =149^\circ 2' 10{,}5"\text{ of } \alpha = 329^\circ 2' 10{,}5"\\5-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 329^\circ 2' 10{,}5"\)
11. \(6-8i\\ r = \sqrt{6^2+(-8)^2} = \sqrt{100} \\ \alpha = tan^{-1}(\frac{-8}{6}) \Leftrightarrow \alpha =126^\circ 52' 11{,}6"\text{ of } \alpha = 306^\circ 52' 11{,}6"\\6-8i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 306^\circ 52' 11{,}6"\)
12. \(9-2i\\ r = \sqrt{9^2+(-2)^2} = \sqrt{85} \\ \alpha = tan^{-1}(\frac{-2}{9}) \Leftrightarrow \alpha =167^\circ 28' 16{,}3"\text{ of } \alpha = 347^\circ 28' 16{,}3"\\9-2i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 347^\circ 28' 16{,}3"\)