Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(10+i\\ r = \sqrt{10^2+1^2} = \sqrt{101} \\ \alpha = tan^{-1}(\frac{1}{10}) \Leftrightarrow \alpha =5^\circ 42' 38{,}1"\text{ of } \alpha = 185^\circ 42' 38{,}1"\\10+i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 5^\circ 42' 38{,}1"\)
2. \(1+2i\\ r = \sqrt{1^2+2^2} = \sqrt{5} \\ \alpha = tan^{-1}(\frac{2}{1}) \Leftrightarrow \alpha =63^\circ 26' 5{,}8"\text{ of } \alpha = 243^\circ 26' 5{,}8"\\1+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 63^\circ 26' 5{,}8"\)
3. \(1-6i\\ r = \sqrt{1^2+(-6)^2} = \sqrt{37} \\ \alpha = tan^{-1}(\frac{-6}{1}) \Leftrightarrow \alpha =99^\circ 27' 44{,}4"\text{ of } \alpha = 279^\circ 27' 44{,}4"\\1-6i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 279^\circ 27' 44{,}4"\)
4. \(5-8i\\ r = \sqrt{5^2+(-8)^2} = \sqrt{89} \\ \alpha = tan^{-1}(\frac{-8}{5}) \Leftrightarrow \alpha =122^\circ 0' 19{,}4"\text{ of } \alpha = 302^\circ 0' 19{,}4"\\5-8i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 302^\circ 0' 19{,}4"\)
5. \(1+3i\\ r = \sqrt{1^2+3^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{3}{1}) \Leftrightarrow \alpha =71^\circ 33' 54{,}2"\text{ of } \alpha = 251^\circ 33' 54{,}2"\\1+3i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 71^\circ 33' 54{,}2"\)
6. \(-4+i\\ r = \sqrt{(-4)^2+1^2} = \sqrt{17} \\ \alpha = tan^{-1}(\frac{1}{-4}) \Leftrightarrow \alpha =165^\circ 57' 49{,}5"\text{ of } \alpha = 345^\circ 57' 49{,}5"\\-4+i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 165^\circ 57' 49{,}5"\)
7. \(-5+4i\\ r = \sqrt{(-5)^2+4^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{4}{-5}) \Leftrightarrow \alpha =141^\circ 20' 24{,}7"\text{ of } \alpha = 321^\circ 20' 24{,}7"\\-5+4i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 141^\circ 20' 24{,}7"\)
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. \(9+6i\\ r = \sqrt{9^2+6^2} = \sqrt{117} \\ \alpha = tan^{-1}(\frac{6}{9}) \Leftrightarrow \alpha =33^\circ 41' 24{,}2"\text{ of } \alpha = 213^\circ 41' 24{,}2"\\9+6i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 33^\circ 41' 24{,}2"\)
10. \(-9+6i\\ r = \sqrt{(-9)^2+6^2} = \sqrt{117} \\ \alpha = tan^{-1}(\frac{6}{-9}) \Leftrightarrow \alpha =146^\circ 18' 35{,}8"\text{ of } \alpha = 326^\circ 18' 35{,}8"\\-9+6i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 146^\circ 18' 35{,}8"\)
11. \(-3-3i\\ r = \sqrt{(-3)^2+(-3)^2} = \sqrt{18} \\ \alpha = tan^{-1}(\frac{-3}{-3}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\-3-3i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 225^\circ \)
12. \(-6+8i\\ r = \sqrt{(-6)^2+8^2} = \sqrt{100} \\ \alpha = tan^{-1}(\frac{8}{-6}) \Leftrightarrow \alpha =126^\circ 52' 11{,}6"\text{ of } \alpha = 306^\circ 52' 11{,}6"\\-6+8i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 126^\circ 52' 11{,}6"\)