Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(3+i\\ r = \sqrt{3^2+1^2} = \sqrt{10} \\ \alpha = tan^{-1}(\frac{1}{3}) \Leftrightarrow \alpha =18^\circ 26' 5{,}8"\text{ of } \alpha = 198^\circ 26' 5{,}8"\\3+i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 18^\circ 26' 5{,}8"\)
2. \(-6+5i\\ r = \sqrt{(-6)^2+5^2} = \sqrt{61} \\ \alpha = tan^{-1}(\frac{5}{-6}) \Leftrightarrow \alpha =140^\circ 11' 39{,}9"\text{ of } \alpha = 320^\circ 11' 39{,}9"\\-6+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 140^\circ 11' 39{,}9"\)
3. \(-6-7i\\ r = \sqrt{(-6)^2+(-7)^2} = \sqrt{85} \\ \alpha = tan^{-1}(\frac{-7}{-6}) \Leftrightarrow \alpha =49^\circ 23' 55{,}3"\text{ of } \alpha = 229^\circ 23' 55{,}3"\\-6-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 229^\circ 23' 55{,}3"\)
4. \(10+8i\\ r = \sqrt{10^2+8^2} = \sqrt{164} \\ \alpha = tan^{-1}(\frac{8}{10}) \Leftrightarrow \alpha =38^\circ 39' 35{,}3"\text{ of } \alpha = 218^\circ 39' 35{,}3"\\10+8i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 38^\circ 39' 35{,}3"\)
5. \(-7-6i\\ r = \sqrt{(-7)^2+(-6)^2} = \sqrt{85} \\ \alpha = tan^{-1}(\frac{-6}{-7}) \Leftrightarrow \alpha =40^\circ 36' 4{,}7"\text{ of } \alpha = 220^\circ 36' 4{,}7"\\-7-6i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 220^\circ 36' 4{,}7"\)
6. \(1-7i\\ r = \sqrt{1^2+(-7)^2} = \sqrt{50} \\ \alpha = tan^{-1}(\frac{-7}{1}) \Leftrightarrow \alpha =98^\circ 7' 48{,}4"\text{ of } \alpha = 278^\circ 7' 48{,}4"\\1-7i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 278^\circ 7' 48{,}4"\)
7. \(-5-4i\\ r = \sqrt{(-5)^2+(-4)^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{-4}{-5}) \Leftrightarrow \alpha =38^\circ 39' 35{,}3"\text{ of } \alpha = 218^\circ 39' 35{,}3"\\-5-4i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 218^\circ 39' 35{,}3"\)
8. \(-8+8i\\ r = \sqrt{(-8)^2+8^2} = \sqrt{128} \\ \alpha = tan^{-1}(\frac{8}{-8}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-8+8i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)
9. \(-4-4i\\ r = \sqrt{(-4)^2+(-4)^2} = \sqrt{32} \\ \alpha = tan^{-1}(\frac{-4}{-4}) \Leftrightarrow \alpha =45^\circ \text{ of } \alpha = 225^\circ \\-4-4i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 225^\circ \)
10. \(6-7i\\ r = \sqrt{6^2+(-7)^2} = \sqrt{85} \\ \alpha = tan^{-1}(\frac{-7}{6}) \Leftrightarrow \alpha =130^\circ 36' 4{,}7"\text{ of } \alpha = 310^\circ 36' 4{,}7"\\6-7i\text{ ligt in kwadrant }4, \alpha \text{ ligt dus tussen }270^\circ \text{ en }360^\circ\\ \alpha = 310^\circ 36' 4{,}7"\)
11. \(-9+5i\\ r = \sqrt{(-9)^2+5^2} = \sqrt{106} \\ \alpha = tan^{-1}(\frac{5}{-9}) \Leftrightarrow \alpha =150^\circ 56' 43{,}4"\text{ of } \alpha = 330^\circ 56' 43{,}4"\\-9+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 150^\circ 56' 43{,}4"\)
12. \(-10+3i\\ r = \sqrt{(-10)^2+3^2} = \sqrt{109} \\ \alpha = tan^{-1}(\frac{3}{-10}) \Leftrightarrow \alpha =163^\circ 18' 2{,}7"\text{ of } \alpha = 343^\circ 18' 2{,}7"\\-10+3i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 163^\circ 18' 2{,}7"\)