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Πολική μορφή του $$$15625 + \frac{719413999 i}{1000000000}$$$
Η είσοδός σας
Βρείτε την πολική μορφή του $$$15625 + \frac{719413999 i}{1000000000}$$$.
Λύση
Η κανονική μορφή του μιγαδικού αριθμού είναι $$$15625 + \frac{719413999 i}{1000000000}$$$.
Για έναν μιγαδικό αριθμό $$$a + b i$$$, η πολική μορφή δίνεται από $$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$, όπου $$$r = \sqrt{a^{2} + b^{2}}$$$ και $$$\theta = \operatorname{atan}{\left(\frac{b}{a} \right)}$$$.
Έχουμε ότι $$$a = 15625$$$ και $$$b = \frac{719413999}{1000000000}$$$.
Άρα, $$$r = \sqrt{15625^{2} + \left(\frac{719413999}{1000000000}\right)^{2}} = \frac{\sqrt{244140625517556501957172001}}{1000000000}.$$$
Επίσης, $$$\theta = \operatorname{atan}{\left(\frac{\frac{719413999}{1000000000}}{15625} \right)} = \operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}.$$$
Επομένως, $$$15625 + \frac{719413999 i}{1000000000} = \frac{\sqrt{244140625517556501957172001}}{1000000000} \left(\cos{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)}\right).$$$
Απάντηση
$$$15625 + \frac{719413999 i}{1000000000} = \frac{\sqrt{244140625517556501957172001}}{1000000000} \left(\cos{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)}\right) = \frac{\sqrt{244140625517556501957172001}}{1000000000} \left(\cos{\left(\left(\frac{180 \operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{\pi}\right)^{\circ} \right)} + i \sin{\left(\left(\frac{180 \operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{\pi}\right)^{\circ} \right)}\right)$$$A