Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(5+i\\ r = \sqrt{5^2+1^2} = \sqrt{26} \\ \alpha = tan^{-1}(\frac{1}{5}) \Leftrightarrow \alpha =11^\circ 18' 35{,}8"\text{ of } \alpha = 191^\circ 18' 35{,}8"\\5+i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 11^\circ 18' 35{,}8"\)
2. \(4-9i\\ r = \sqrt{4^2+(-9)^2} = \sqrt{97} \\ \alpha = tan^{-1}(\frac{-9}{4}) \Leftrightarrow \alpha =113^\circ 57' 45"\text{ of } \alpha = 293^\circ 57' 45"\\4-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 293^\circ 57' 45"\)
3. \(-10-6i\\ r = \sqrt{(-10)^2+(-6)^2} = \sqrt{136} \\ \alpha = tan^{-1}(\frac{-6}{-10}) \Leftrightarrow \alpha =30^\circ 57' 49{,}5"\text{ of } \alpha = 210^\circ 57' 49{,}5"\\-10-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 210^\circ 57' 49{,}5"\)
4. \(-8+6i\\ r = \sqrt{(-8)^2+6^2} = \sqrt{100} \\ \alpha = tan^{-1}(\frac{6}{-8}) \Leftrightarrow \alpha =143^\circ 7' 48{,}4"\text{ of } \alpha = 323^\circ 7' 48{,}4"\\-8+6i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 143^\circ 7' 48{,}4"\)
5. \(1\\ \text{ Dit complex getal ligt op het positief gedeelte van de x-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }1\\\alpha = 0 ^\circ \\\)
6. \(4-8i\\ r = \sqrt{4^2+(-8)^2} = \sqrt{80} \\ \alpha = tan^{-1}(\frac{-8}{4}) \Leftrightarrow \alpha =116^\circ 33' 54{,}2"\text{ of } \alpha = 296^\circ 33' 54{,}2"\\4-8i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 296^\circ 33' 54{,}2"\)
7. \(-9-8i\\ r = \sqrt{(-9)^2+(-8)^2} = \sqrt{145} \\ \alpha = tan^{-1}(\frac{-8}{-9}) \Leftrightarrow \alpha =41^\circ 38' 0{,}7"\text{ of } \alpha = 221^\circ 38' 0{,}7"\\-9-8i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 221^\circ 38' 0{,}7"\)
8. \(-1-3i\\ r = \sqrt{(-1)^2+(-3)^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{-3}{-1}) \Leftrightarrow \alpha =71^\circ 33' 54{,}2"\text{ of } \alpha = 251^\circ 33' 54{,}2"\\-1-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 251^\circ 33' 54{,}2"\)
9. \(-7-9i\\ r = \sqrt{(-7)^2+(-9)^2} = \sqrt{130} \\ \alpha = tan^{-1}(\frac{-9}{-7}) \Leftrightarrow \alpha =52^\circ 7' 30{,}1"\text{ of } \alpha = 232^\circ 7' 30{,}1"\\-7-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 232^\circ 7' 30{,}1"\)
10. \(6+10i\\ r = \sqrt{6^2+10^2} = \sqrt{136} \\ \alpha = tan^{-1}(\frac{10}{6}) \Leftrightarrow \alpha =59^\circ 2' 10{,}5"\text{ of } \alpha = 239^\circ 2' 10{,}5"\\6+10i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 59^\circ 2' 10{,}5"\)
11. \(2+2i\\ r = \sqrt{2^2+2^2} = \sqrt{8} \\ \alpha = tan^{-1}(\frac{2}{2}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\2+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
12. \(-2+3i\\ r = \sqrt{(-2)^2+3^2} = \sqrt{13} \\ \alpha = tan^{-1}(\frac{3}{-2}) \Leftrightarrow \alpha =123^\circ 41' 24{,}2"\text{ of } \alpha = 303^\circ 41' 24{,}2"\\-2+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 123^\circ 41' 24{,}2"\)