Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(3+6i\\ r = \sqrt{3^2+6^2} = \sqrt{45} \\ \alpha = tan^{-1}(\frac{6}{3}) \Leftrightarrow \alpha =63^\circ 26' 5{,}8"\text{ of } \alpha = 243^\circ 26' 5{,}8"\\3+6i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 63^\circ 26' 5{,}8"\)
2. \(3-i\\ r = \sqrt{3^2+(-1)^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{-1}{3}) \Leftrightarrow \alpha =161^\circ 33' 54{,}2"\text{ of } \alpha = 341^\circ 33' 54{,}2"\\3-i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 341^\circ 33' 54{,}2"\)
3. \(5-7i\\ r = \sqrt{5^2+(-7)^2} = \sqrt{74} \\ \alpha = tan^{-1}(\frac{-7}{5}) \Leftrightarrow \alpha =125^\circ 32' 15{,}6"\text{ of } \alpha = 305^\circ 32' 15{,}6"\\5-7i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 305^\circ 32' 15{,}6"\)
4. \(1-2i\\ r = \sqrt{1^2+(-2)^2} = \sqrt{5} \\ \alpha = tan^{-1}(\frac{-2}{1}) \Leftrightarrow \alpha =116^\circ 33' 54{,}2"\text{ of } \alpha = 296^\circ 33' 54{,}2"\\1-2i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 296^\circ 33' 54{,}2"\)
5. \(6+6i\\ r = \sqrt{6^2+6^2} = \sqrt{72} \\ \alpha = tan^{-1}(\frac{6}{6}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\6+6i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 45^\circ \)
6. \(-8-7i\\ r = \sqrt{(-8)^2+(-7)^2} = \sqrt{113} \\ \alpha = tan^{-1}(\frac{-7}{-8}) \Leftrightarrow \alpha =41^\circ 11' 9{,}3"\text{ of } \alpha = 221^\circ 11' 9{,}3"\\-8-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 221^\circ 11' 9{,}3"\)
7. \(3+5i\\ r = \sqrt{3^2+5^2} = \sqrt{34} \\ \alpha = tan^{-1}(\frac{5}{3}) \Leftrightarrow \alpha =59^\circ 2' 10{,}5"\text{ of } \alpha = 239^\circ 2' 10{,}5"\\3+5i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 59^\circ 2' 10{,}5"\)
8. \(10+9i\\ r = \sqrt{10^2+9^2} = \sqrt{181} \\ \alpha = tan^{-1}(\frac{9}{10}) \Leftrightarrow \alpha =41^\circ 59' 14"\text{ of } \alpha = 221^\circ 59' 14"\\10+9i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 41^\circ 59' 14"\)
9. \(4+i\\ r = \sqrt{4^2+1^2} = \sqrt{17} \\ \alpha = tan^{-1}(\frac{1}{4}) \Leftrightarrow \alpha =14^\circ 2' 10{,}5"\text{ of } \alpha = 194^\circ 2' 10{,}5"\\4+i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 14^\circ 2' 10{,}5"\)
10. \(1+3i\\ r = \sqrt{1^2+3^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{3}{1}) \Leftrightarrow \alpha =71^\circ 33' 54{,}2"\text{ of } \alpha = 251^\circ 33' 54{,}2"\\1+3i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 71^\circ 33' 54{,}2"\)
11. \(7+3i\\ r = \sqrt{7^2+3^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{3}{7}) \Leftrightarrow \alpha =23^\circ 11' 54{,}9"\text{ of } \alpha = 203^\circ 11' 54{,}9"\\7+3i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 23^\circ 11' 54{,}9"\)
12. \(-10+8i\\ r = \sqrt{(-10)^2+8^2} = \sqrt{164} \\ \alpha = tan^{-1}(\frac{8}{-10}) \Leftrightarrow \alpha =141^\circ 20' 24{,}7"\text{ of } \alpha = 321^\circ 20' 24{,}7"\\-10+8i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 141^\circ 20' 24{,}7"\)