Bepaal modulus en argument

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Bepaal modulus en argument

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

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Bepaal modulus en argument

Verbetersleutel

1. \(2+7i\\ r = \sqrt{2^2+7^2} = \sqrt{53} \\ \alpha = tan^{-1}(\frac{7}{2}) \Leftrightarrow \alpha =74^\circ 3' 16{,}6"\text{ of } \alpha = 254^\circ 3' 16{,}6"\\2+7i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 74^\circ 3' 16{,}6"\)
2. \(-2-i\\ r = \sqrt{(-2)^2+(-1)^2} = \sqrt{5} \\ \alpha = tan^{-1}(\frac{-1}{-2}) \Leftrightarrow \alpha =26^\circ 33' 54{,}2"\text{ of } \alpha = 206^\circ 33' 54{,}2"\\-2-i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 206^\circ 33' 54{,}2"\)
3. \(10+7i\\ r = \sqrt{10^2+7^2} = \sqrt{149} \\ \alpha = tan^{-1}(\frac{7}{10}) \Leftrightarrow \alpha =34^\circ 59' 31{,}3"\text{ of } \alpha = 214^\circ 59' 31{,}3"\\10+7i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 34^\circ 59' 31{,}3"\)
4. \(-10-9i\\ r = \sqrt{(-10)^2+(-9)^2} = \sqrt{181} \\ \alpha = tan^{-1}(\frac{-9}{-10}) \Leftrightarrow \alpha =41^\circ 59' 14"\text{ of } \alpha = 221^\circ 59' 14"\\-10-9i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 221^\circ 59' 14"\)
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. \(-10+10i\\ r = \sqrt{(-10)^2+10^2} = \sqrt{200} \\ \alpha = tan^{-1}(\frac{10}{-10}) \Leftrightarrow \alpha =135^\circ \text{ of } \alpha = 315^\circ \\-10+10i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 135^\circ \)
7. \(-8-5i\\ r = \sqrt{(-8)^2+(-5)^2} = \sqrt{89} \\ \alpha = tan^{-1}(\frac{-5}{-8}) \Leftrightarrow \alpha =32^\circ 0' 19{,}4"\text{ of } \alpha = 212^\circ 0' 19{,}4"\\-8-5i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 212^\circ 0' 19{,}4"\)
8. \(-3-7i\\ r = \sqrt{(-3)^2+(-7)^2} = \sqrt{58} \\ \alpha = tan^{-1}(\frac{-7}{-3}) \Leftrightarrow \alpha =66^\circ 48' 5{,}1"\text{ of } \alpha = 246^\circ 48' 5{,}1"\\-3-7i\text{ ligt in kwadrant }3, \alpha \text{ ligt dus tussen }180^\circ \text{ en }270^\circ\\ \alpha = 246^\circ 48' 5{,}1"\)
9. \(-8+5i\\ r = \sqrt{(-8)^2+5^2} = \sqrt{89} \\ \alpha = tan^{-1}(\frac{5}{-8}) \Leftrightarrow \alpha =147^\circ 59' 40{,}6"\text{ of } \alpha = 327^\circ 59' 40{,}6"\\-8+5i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 147^\circ 59' 40{,}6"\)
10. \(1+9i\\ r = \sqrt{1^2+9^2} = \sqrt{82} \\ \alpha = tan^{-1}(\frac{9}{1}) \Leftrightarrow \alpha =83^\circ 39' 35{,}3"\text{ of } \alpha = 263^\circ 39' 35{,}3"\\1+9i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 83^\circ 39' 35{,}3"\)
11. \(-7+8i\\ r = \sqrt{(-7)^2+8^2} = \sqrt{113} \\ \alpha = tan^{-1}(\frac{8}{-7}) \Leftrightarrow \alpha =131^\circ 11' 9{,}3"\text{ of } \alpha = 311^\circ 11' 9{,}3"\\-7+8i\text{ ligt in kwadrant }2, \alpha \text{ ligt dus tussen }90^\circ \text{ en }180^\circ\\ \alpha = 131^\circ 11' 9{,}3"\)
12. \(5+4i\\ r = \sqrt{5^2+4^2} = \sqrt{41} \\ \alpha = tan^{-1}(\frac{4}{5}) \Leftrightarrow \alpha =38^\circ 39' 35{,}3"\text{ of } \alpha = 218^\circ 39' 35{,}3"\\5+4i\text{ ligt in kwadrant }1, \alpha \text{ ligt dus tussen }0^\circ \text{ en }90^\circ\\ \alpha = 38^\circ 39' 35{,}3"\)