z^4=3+2i

解

z^{4} = 3 + 2 i

z = 813 cos (\frac{arctan ( \frac{2}{3} )}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} )}{4}), z = 813 cos (\frac{arctan ( \frac{2}{3} ) + 2 π}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} ) + 2 π}{4}), z = 813 cos (\frac{arctan ( \frac{2}{3} ) + 4 π}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} ) + 4 π}{4}), z = 813 cos (\frac{arctan ( \frac{2}{3} ) + 6 π}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} ) + 6 π}{4})

解答ステップ

z^{4} = 3 + 2 i

z^{n} = a

の場合, 解は

z_{k} = n ∣ a ∣ (cos (\frac{ar g ( a ) + 2 kπ}{n}) + i sin (\frac{ar g ( a ) + 2 kπ}{n})),

k = 0, 1, \dots, n - 1

以下のため:

n = 4, a = 3 + 2 i

∣ a ∣ = 13

ar g (a) = arctan (\frac{2}{3})

z = 413 (cos (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 0 π}{4}) + i sin (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 0 π}{4})), z = 413 (cos (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 1 π}{4}) + i sin (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 1 π}{4})), z = 413 (cos (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 2 π}{4}) + i sin (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 2 π}{4})), z = 413 (cos (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 3 π}{4}) + i sin (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 3 π}{4}))

簡素化

413 (cos (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 0 π}{4}) + i sin (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 0 π}{4})) : 813 cos (\frac{arctan ( \frac{2}{3} )}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} )}{4})

簡素化

413 (cos (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 1 π}{4}) + i sin (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 1 π}{4})) : 813 cos (\frac{arctan ( \frac{2}{3} ) + 2 π}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} ) + 2 π}{4})

簡素化

413 (cos (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 2 π}{4}) + i sin (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 2 π}{4})) : 813 cos (\frac{arctan ( \frac{2}{3} ) + 4 π}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} ) + 4 π}{4})

簡素化

413 (cos (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 3 π}{4}) + i sin (\frac{arctan ( \frac{2}{3} ) + 2 \cdot 3 π}{4})) : 813 cos (\frac{arctan ( \frac{2}{3} ) + 6 π}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} ) + 6 π}{4})

z = 813 cos (\frac{arctan ( \frac{2}{3} )}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} )}{4}), z = 813 cos (\frac{arctan ( \frac{2}{3} ) + 2 π}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} ) + 2 π}{4}), z = 813 cos (\frac{arctan ( \frac{2}{3} ) + 4 π}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} ) + 4 π}{4}), z = 813 cos (\frac{arctan ( \frac{2}{3} ) + 6 π}{4}) + 813 i sin (\frac{arctan ( \frac{2}{3} ) + 6 π}{4})