Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(3+8i\\ r = \sqrt{3^2+8^2} = \sqrt{73} \\ \alpha = tan^{-1}(\frac{8}{3}) \Leftrightarrow \alpha =69^\circ 26' 38{,}2"\text{ of } \alpha = 249^\circ 26' 38{,}2"\\3+8i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 69^\circ 26' 38{,}2"\)
2. \(1+4i\\ r = \sqrt{1^2+4^2} = \sqrt{17} \\ \alpha = tan^{-1}(\frac{4}{1}) \Leftrightarrow \alpha =75^\circ 57' 49{,}5"\text{ of } \alpha = 255^\circ 57' 49{,}5"\\1+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 75^\circ 57' 49{,}5"\)
3. \(-4+3i\\ r = \sqrt{(-4)^2+3^2} = \sqrt{25} \\ \alpha = tan^{-1}(\frac{3}{-4}) \Leftrightarrow \alpha =143^\circ 7' 48{,}4"\text{ of } \alpha = 323^\circ 7' 48{,}4"\\-4+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 143^\circ 7' 48{,}4"\)
4. \(5+7i\\ r = \sqrt{5^2+7^2} = \sqrt{74} \\ \alpha = tan^{-1}(\frac{7}{5}) \Leftrightarrow \alpha =54^\circ 27' 44{,}4"\text{ of } \alpha = 234^\circ 27' 44{,}4"\\5+7i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 54^\circ 27' 44{,}4"\)
5. \(3-7i\\ r = \sqrt{3^2+(-7)^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{-7}{3}) \Leftrightarrow \alpha =113^\circ 11' 54{,}9"\text{ of } \alpha = 293^\circ 11' 54{,}9"\\3-7i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 293^\circ 11' 54{,}9"\)
6. \(1-2i\\ r = \sqrt{1^2+(-2)^2} = \sqrt{5} \\ \alpha = tan^{-1}(\frac{-2}{1}) \Leftrightarrow \alpha =116^\circ 33' 54{,}2"\text{ of } \alpha = 296^\circ 33' 54{,}2"\\1-2i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 296^\circ 33' 54{,}2"\)
7. \(-3+9i\\ r = \sqrt{(-3)^2+9^2} = \sqrt{90} \\ \alpha = tan^{-1}(\frac{9}{-3}) \Leftrightarrow \alpha =108^\circ 26' 5{,}8"\text{ of } \alpha = 288^\circ 26' 5{,}8"\\-3+9i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 108^\circ 26' 5{,}8"\)
8. \(2+7i\\ r = \sqrt{2^2+7^2} = \sqrt{53} \\ \alpha = tan^{-1}(\frac{7}{2}) \Leftrightarrow \alpha =74^\circ 3' 16{,}6"\text{ of } \alpha = 254^\circ 3' 16{,}6"\\2+7i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 74^\circ 3' 16{,}6"\)
9. \(-3+i\\ r = \sqrt{(-3)^2+1^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{1}{-3}) \Leftrightarrow \alpha =161^\circ 33' 54{,}2"\text{ of } \alpha = 341^\circ 33' 54{,}2"\\-3+i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 161^\circ 33' 54{,}2"\)
10. \(-2+6i\\ r = \sqrt{(-2)^2+6^2} = \sqrt{40} \\ \alpha = tan^{-1}(\frac{6}{-2}) \Leftrightarrow \alpha =108^\circ 26' 5{,}8"\text{ of } \alpha = 288^\circ 26' 5{,}8"\\-2+6i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 108^\circ 26' 5{,}8"\)
11. \(-5-8i\\ r = \sqrt{(-5)^2+(-8)^2} = \sqrt{89} \\ \alpha = tan^{-1}(\frac{-8}{-5}) \Leftrightarrow \alpha =57^\circ 59' 40{,}6"\text{ of } \alpha = 237^\circ 59' 40{,}6"\\-5-8i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 237^\circ 59' 40{,}6"\)
12. \(1\\ \text{ Dit complex getal ligt op het positief gedeelte van de x-as. We hebben geen berekeningen nodig om r of } \alpha \text{ te berekenen.} \\\text{r = }1\\\alpha = 0 ^\circ \\\)