Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-5-6i\\ r = \sqrt{(-5)^2+(-6)^2} = \sqrt{61} \\ \alpha = tan^{-1}(\frac{-6}{-5}) \Leftrightarrow \alpha =50^\circ 11' 39{,}9"\text{ of } \alpha = 230^\circ 11' 39{,}9"\\-5-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 230^\circ 11' 39{,}9"\)
2. \(10+3i\\ r = \sqrt{10^2+3^2} = \sqrt{109} \\ \alpha = tan^{-1}(\frac{3}{10}) \Leftrightarrow \alpha =16^\circ 41' 57{,}3"\text{ of } \alpha = 196^\circ 41' 57{,}3"\\10+3i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 16^\circ 41' 57{,}3"\)
3. \(4-3i\\ r = \sqrt{4^2+(-3)^2} = \sqrt{25} \\ \alpha = tan^{-1}(\frac{-3}{4}) \Leftrightarrow \alpha =143^\circ 7' 48{,}4"\text{ of } \alpha = 323^\circ 7' 48{,}4"\\4-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 323^\circ 7' 48{,}4"\)
4. \(-5-i\\ r = \sqrt{(-5)^2+(-1)^2} = \sqrt{26} \\ \alpha = tan^{-1}(\frac{-1}{-5}) \Leftrightarrow \alpha =11^\circ 18' 35{,}8"\text{ of } \alpha = 191^\circ 18' 35{,}8"\\-5-i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 191^\circ 18' 35{,}8"\)
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. \(10-10i\\ r = \sqrt{10^2+(-10)^2} = \sqrt{200} \\ \alpha = tan^{-1}(\frac{-10}{10}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\10-10i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 315^\circ \)
7. \(7-i\\ r = \sqrt{7^2+(-1)^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{-1}{7}) \Leftrightarrow \alpha =171^\circ 52' 11{,}6"\text{ of } \alpha = 351^\circ 52' 11{,}6"\\7-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 351^\circ 52' 11{,}6"\)
8. \(-9+8i\\ r = \sqrt{(-9)^2+8^2} = \sqrt{145} \\ \alpha = tan^{-1}(\frac{8}{-9}) \Leftrightarrow \alpha =138^\circ 21' 59{,}3"\text{ of } \alpha = 318^\circ 21' 59{,}3"\\-9+8i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 138^\circ 21' 59{,}3"\)
9. \(-8-9i\\ r = \sqrt{(-8)^2+(-9)^2} = \sqrt{145} \\ \alpha = tan^{-1}(\frac{-9}{-8}) \Leftrightarrow \alpha =48^\circ 21' 59{,}3"\text{ of } \alpha = 228^\circ 21' 59{,}3"\\-8-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 228^\circ 21' 59{,}3"\)
10. \(4+2i\\ r = \sqrt{4^2+2^2} = \sqrt{20} \\ \alpha = tan^{-1}(\frac{2}{4}) \Leftrightarrow \alpha =26^\circ 33' 54{,}2"\text{ of } \alpha = 206^\circ 33' 54{,}2"\\4+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 26^\circ 33' 54{,}2"\)
11. \(7+6i\\ r = \sqrt{7^2+6^2} = \sqrt{85} \\ \alpha = tan^{-1}(\frac{6}{7}) \Leftrightarrow \alpha =40^\circ 36' 4{,}7"\text{ of } \alpha = 220^\circ 36' 4{,}7"\\7+6i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 40^\circ 36' 4{,}7"\)
12. \(-3i\\ \text{ Dit complex getal ligt op het negatief gedeelte van de y-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }3\\\alpha = 270 ^\circ \\\)