Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-7-5i\\ r = \sqrt{(-7)^2+(-5)^2} = \sqrt{74} \\ \alpha = tan^{-1}(\frac{-5}{-7}) \Leftrightarrow \alpha =35^\circ 32' 15{,}6"\text{ of } \alpha = 215^\circ 32' 15{,}6"\\-7-5i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 215^\circ 32' 15{,}6"\)
2. \(-2+10i\\ r = \sqrt{(-2)^2+10^2} = \sqrt{104} \\ \alpha = tan^{-1}(\frac{10}{-2}) \Leftrightarrow \alpha =101^\circ 18' 35{,}8"\text{ of } \alpha = 281^\circ 18' 35{,}8"\\-2+10i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 101^\circ 18' 35{,}8"\)
3. \(-8\\ \text{ Dit complex getal ligt op het negatief gedeelte van de x-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }8\\\alpha = 180 ^\circ \\\)
4. \(-7+i\\ r = \sqrt{(-7)^2+1^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{1}{-7}) \Leftrightarrow \alpha =171^\circ 52' 11{,}6"\text{ of } \alpha = 351^\circ 52' 11{,}6"\\-7+i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 171^\circ 52' 11{,}6"\)
5. \(-10-10i\\ r = \sqrt{(-10)^2+(-10)^2} = \sqrt{200} \\ \alpha = tan^{-1}(\frac{-10}{-10}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\-10-10i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 225^\circ \)
6. \(4+2i\\ r = \sqrt{4^2+2^2} = \sqrt{20} \\ \alpha = tan^{-1}(\frac{2}{4}) \Leftrightarrow \alpha =26^\circ 33' 54{,}2"\text{ of } \alpha = 206^\circ 33' 54{,}2"\\4+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 26^\circ 33' 54{,}2"\)
7. \(-3-7i\\ r = \sqrt{(-3)^2+(-7)^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{-7}{-3}) \Leftrightarrow \alpha =66^\circ 48' 5{,}1"\text{ of } \alpha = 246^\circ 48' 5{,}1"\\-3-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 246^\circ 48' 5{,}1"\)
8. \(3-10i\\ r = \sqrt{3^2+(-10)^2} = \sqrt{109} \\ \alpha = tan^{-1}(\frac{-10}{3}) \Leftrightarrow \alpha =106^\circ 41' 57{,}3"\text{ of } \alpha = 286^\circ 41' 57{,}3"\\3-10i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 286^\circ 41' 57{,}3"\)
9. \(9-10i\\ r = \sqrt{9^2+(-10)^2} = \sqrt{181} \\ \alpha = tan^{-1}(\frac{-10}{9}) \Leftrightarrow \alpha =131^\circ 59' 14"\text{ of } \alpha = 311^\circ 59' 14"\\9-10i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 311^\circ 59' 14"\)
10. \(7+3i\\ r = \sqrt{7^2+3^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{3}{7}) \Leftrightarrow \alpha =23^\circ 11' 54{,}9"\text{ of } \alpha = 203^\circ 11' 54{,}9"\\7+3i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 23^\circ 11' 54{,}9"\)
11. \(2+8i\\ r = \sqrt{2^2+8^2} = \sqrt{68} \\ \alpha = tan^{-1}(\frac{8}{2}) \Leftrightarrow \alpha =75^\circ 57' 49{,}5"\text{ of } \alpha = 255^\circ 57' 49{,}5"\\2+8i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 75^\circ 57' 49{,}5"\)
12. \(6-i\\ r = \sqrt{6^2+(-1)^2} = \sqrt{37} \\ \alpha = tan^{-1}(\frac{-1}{6}) \Leftrightarrow \alpha =170^\circ 32' 15{,}6"\text{ of } \alpha = 350^\circ 32' 15{,}6"\\6-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 350^\circ 32' 15{,}6"\)