Polaire vorm van $$$15625 + \frac{719413999 i}{1000000000}$$$

Bepaal de poolvorm van $$$15625 + \frac{719413999 i}{1000000000}$$$.

De standaardvorm van het complexe getal is $$$15625 + \frac{719413999 i}{1000000000}$$$.

Voor een complex getal $$$a + b i$$$ wordt de polaire vorm gegeven door $$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$, waarbij $$$r = \sqrt{a^{2} + b^{2}}$$$ en $$$\theta = \operatorname{atan}{\left(\frac{b}{a} \right)}$$$.

We hebben dat $$$a = 15625$$$ en $$$b = \frac{719413999}{1000000000}$$$.

Dus, $$$r = \sqrt{15625^{2} + \left(\frac{719413999}{1000000000}\right)^{2}} = \frac{\sqrt{244140625517556501957172001}}{1000000000}.$$$

Bovendien geldt $$$\theta = \operatorname{atan}{\left(\frac{\frac{719413999}{1000000000}}{15625} \right)} = \operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}.$$$

Daarom geldt $$$15625 + \frac{719413999 i}{1000000000} = \frac{\sqrt{244140625517556501957172001}}{1000000000} \left(\cos{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)}\right).$$$

$$$15625 + \frac{719413999 i}{1000000000} = \frac{\sqrt{244140625517556501957172001}}{1000000000} \left(\cos{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)}\right) = \frac{\sqrt{244140625517556501957172001}}{1000000000} \left(\cos{\left(\left(\frac{180 \operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{\pi}\right)^{\circ} \right)} + i \sin{\left(\left(\frac{180 \operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{\pi}\right)^{\circ} \right)}\right)$$$A