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$$$15625 + \frac{719413999 i}{1000000000}$$$ 的極座標形式
您的輸入
求$$$15625 + \frac{719413999 i}{1000000000}$$$的極座標形式。
解答
該複數的標準形式為 $$$15625 + \frac{719413999 i}{1000000000}$$$。
對於複數 $$$a + b i$$$，其極座標形式表示為 $$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$，其中 $$$r = \sqrt{a^{2} + b^{2}}$$$ 和 $$$\theta = \operatorname{atan}{\left(\frac{b}{a} \right)}$$$。
我們有 $$$a = 15625$$$ 與 $$$b = \frac{719413999}{1000000000}$$$。
因此，$$$r = \sqrt{15625^{2} + \left(\frac{719413999}{1000000000}\right)^{2}} = \frac{\sqrt{244140625517556501957172001}}{1000000000}$$$。
此外，$$$\theta = \operatorname{atan}{\left(\frac{\frac{719413999}{1000000000}}{15625} \right)} = \operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}$$$。
因此，$$$15625 + \frac{719413999 i}{1000000000} = \frac{\sqrt{244140625517556501957172001}}{1000000000} \left(\cos{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)}\right)$$$。
答案
$$$15625 + \frac{719413999 i}{1000000000} = \frac{\sqrt{244140625517556501957172001}}{1000000000} \left(\cos{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)} \right)}\right) = \frac{\sqrt{244140625517556501957172001}}{1000000000} \left(\cos{\left(\left(\frac{180 \operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{\pi}\right)^{\circ} \right)} + i \sin{\left(\left(\frac{180 \operatorname{atan}{\left(\frac{719413999}{15625000000000} \right)}}{\pi}\right)^{\circ} \right)}\right)$$$A