Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(-4-4i\\ r = \sqrt{(-4)^2+(-4)^2} = \sqrt{32} \\ \alpha = tan^{-1}(\frac{-4}{-4}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\-4-4i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 225^\circ \)
2. \(-8-i\\ r = \sqrt{(-8)^2+(-1)^2} = \sqrt{65} \\ \alpha = tan^{-1}(\frac{-1}{-8}) \Leftrightarrow \alpha =7^\circ 7' 30{,}1"\text{ of } \alpha = 187^\circ 7' 30{,}1"\\-8-i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 187^\circ 7' 30{,}1"\)
3. \(-4+8i\\ r = \sqrt{(-4)^2+8^2} = \sqrt{80} \\ \alpha = tan^{-1}(\frac{8}{-4}) \Leftrightarrow \alpha =116^\circ 33' 54{,}2"\text{ of } \alpha = 296^\circ 33' 54{,}2"\\-4+8i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 116^\circ 33' 54{,}2"\)
4. \(6+7i\\ r = \sqrt{6^2+7^2} = \sqrt{85} \\ \alpha = tan^{-1}(\frac{7}{6}) \Leftrightarrow \alpha =49^\circ 23' 55{,}3"\text{ of } \alpha = 229^\circ 23' 55{,}3"\\6+7i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 49^\circ 23' 55{,}3"\)
5. \(10-3i\\ r = \sqrt{10^2+(-3)^2} = \sqrt{109} \\ \alpha = tan^{-1}(\frac{-3}{10}) \Leftrightarrow \alpha =163^\circ 18' 2{,}7"\text{ of } \alpha = 343^\circ 18' 2{,}7"\\10-3i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 343^\circ 18' 2{,}7"\)
6. \(4-9i\\ r = \sqrt{4^2+(-9)^2} = \sqrt{97} \\ \alpha = tan^{-1}(\frac{-9}{4}) \Leftrightarrow \alpha =113^\circ 57' 45"\text{ of } \alpha = 293^\circ 57' 45"\\4-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 293^\circ 57' 45"\)
7. \(1+2i\\ r = \sqrt{1^2+2^2} = \sqrt{5} \\ \alpha = tan^{-1}(\frac{2}{1}) \Leftrightarrow \alpha =63^\circ 26' 5{,}8"\text{ of } \alpha = 243^\circ 26' 5{,}8"\\1+2i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 63^\circ 26' 5{,}8"\)
8. \(-8-9i\\ r = \sqrt{(-8)^2+(-9)^2} = \sqrt{145} \\ \alpha = tan^{-1}(\frac{-9}{-8}) \Leftrightarrow \alpha =48^\circ 21' 59{,}3"\text{ of } \alpha = 228^\circ 21' 59{,}3"\\-8-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 228^\circ 21' 59{,}3"\)
9. \(-3-6i\\ r = \sqrt{(-3)^2+(-6)^2} = \sqrt{45} \\ \alpha = tan^{-1}(\frac{-6}{-3}) \Leftrightarrow \alpha =63^\circ 26' 5{,}8"\text{ of } \alpha = 243^\circ 26' 5{,}8"\\-3-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 243^\circ 26' 5{,}8"\)
10. \(-2+9i\\ r = \sqrt{(-2)^2+9^2} = \sqrt{85} \\ \alpha = tan^{-1}(\frac{9}{-2}) \Leftrightarrow \alpha =102^\circ 31' 43{,}7"\text{ of } \alpha = 282^\circ 31' 43{,}7"\\-2+9i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 102^\circ 31' 43{,}7"\)
11. \(5-9i\\ r = \sqrt{5^2+(-9)^2} = \sqrt{106} \\ \alpha = tan^{-1}(\frac{-9}{5}) \Leftrightarrow \alpha =119^\circ 3' 16{,}6"\text{ of } \alpha = 299^\circ 3' 16{,}6"\\5-9i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 299^\circ 3' 16{,}6"\)
12. \(-7-8i\\ r = \sqrt{(-7)^2+(-8)^2} = \sqrt{113} \\ \alpha = tan^{-1}(\frac{-8}{-7}) \Leftrightarrow \alpha =48^\circ 48' 50{,}7"\text{ of } \alpha = 228^\circ 48' 50{,}7"\\-7-8i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 228^\circ 48' 50{,}7"\)