Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-9-i\\ r = \sqrt{(-9)^2+(-1)^2} = \sqrt{82} \\ \alpha = tan^{-1}(\frac{-1}{-9}) \Leftrightarrow \alpha =6^\circ 20' 24{,}7"\text{ of } \alpha = 186^\circ 20' 24{,}7"\\-9-i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 186^\circ 20' 24{,}7"\)
2. \(5-9i\\ r = \sqrt{5^2+(-9)^2} = \sqrt{106} \\ \alpha = tan^{-1}(\frac{-9}{5}) \Leftrightarrow \alpha =119^\circ 3' 16{,}6"\text{ of } \alpha = 299^\circ 3' 16{,}6"\\5-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 299^\circ 3' 16{,}6"\)
3. \(2-i\\ r = \sqrt{2^2+(-1)^2} = \sqrt{5} \\ \alpha = tan^{-1}(\frac{-1}{2}) \Leftrightarrow \alpha =153^\circ 26' 5{,}8"\text{ of } \alpha = 333^\circ 26' 5{,}8"\\2-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 333^\circ 26' 5{,}8"\)
4. \(10-9i\\ r = \sqrt{10^2+(-9)^2} = \sqrt{181} \\ \alpha = tan^{-1}(\frac{-9}{10}) \Leftrightarrow \alpha =138^\circ 0' 46"\text{ of } \alpha = 318^\circ 0' 46"\\10-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 318^\circ 0' 46"\)
5. \(-5-7i\\ r = \sqrt{(-5)^2+(-7)^2} = \sqrt{74} \\ \alpha = tan^{-1}(\frac{-7}{-5}) \Leftrightarrow \alpha =54^\circ 27' 44{,}4"\text{ of } \alpha = 234^\circ 27' 44{,}4"\\-5-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 234^\circ 27' 44{,}4"\)
6. \(10+9i\\ r = \sqrt{10^2+9^2} = \sqrt{181} \\ \alpha = tan^{-1}(\frac{9}{10}) \Leftrightarrow \alpha =41^\circ 59' 14"\text{ of } \alpha = 221^\circ 59' 14"\\10+9i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 41^\circ 59' 14"\)
7. \(-9-3i\\ r = \sqrt{(-9)^2+(-3)^2} = \sqrt{90} \\ \alpha = tan^{-1}(\frac{-3}{-9}) \Leftrightarrow \alpha =18^\circ 26' 5{,}8"\text{ of } \alpha = 198^\circ 26' 5{,}8"\\-9-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 198^\circ 26' 5{,}8"\)
8. \(2-7i\\ r = \sqrt{2^2+(-7)^2} = \sqrt{53} \\ \alpha = tan^{-1}(\frac{-7}{2}) \Leftrightarrow \alpha =105^\circ 56' 43{,}4"\text{ of } \alpha = 285^\circ 56' 43{,}4"\\2-7i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 285^\circ 56' 43{,}4"\)
9. \(3+4i\\ r = \sqrt{3^2+4^2} = \sqrt{25} \\ \alpha = tan^{-1}(\frac{4}{3}) \Leftrightarrow \alpha =53^\circ 7' 48{,}4"\text{ of } \alpha = 233^\circ 7' 48{,}4"\\3+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 53^\circ 7' 48{,}4"\)
10. \(-4+3i\\ r = \sqrt{(-4)^2+3^2} = \sqrt{25} \\ \alpha = tan^{-1}(\frac{3}{-4}) \Leftrightarrow \alpha =143^\circ 7' 48{,}4"\text{ of } \alpha = 323^\circ 7' 48{,}4"\\-4+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 143^\circ 7' 48{,}4"\)
11. \(8\\ \text{ Dit complex getal ligt op het positief gedeelte van de x-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }8\\\alpha = 0 ^\circ \\\)
12. \(2-8i\\ r = \sqrt{2^2+(-8)^2} = \sqrt{68} \\ \alpha = tan^{-1}(\frac{-8}{2}) \Leftrightarrow \alpha =104^\circ 2' 10{,}5"\text{ of } \alpha = 284^\circ 2' 10{,}5"\\2-8i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 284^\circ 2' 10{,}5"\)