z^3=16+88i

解

z^{3} = 16 + 88 i

z = 25 cos (\frac{arctan ( \frac{11}{2} )}{3}) + 25 i sin (\frac{arctan ( \frac{11}{2} )}{3}), z = 25 cos (\frac{arctan ( \frac{11}{2} ) + 2 π}{3}) + 25 i sin (\frac{arctan ( \frac{11}{2} ) + 2 π}{3}), z = 25 cos (\frac{arctan ( \frac{11}{2} ) + 4 π}{3}) + 25 i sin (\frac{arctan ( \frac{11}{2} ) + 4 π}{3})

解答ステップ

z^{3} = 16 + 88 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 = 3, a = 16 + 88 i

∣ a ∣ = 405

ar g (a) = arctan (\frac{11}{2})

z = 3405 (cos (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 0 π}{3}) + i sin (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 0 π}{3})), z = 3405 (cos (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 1 π}{3}) + i sin (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 1 π}{3})), z = 3405 (cos (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 2 π}{3}) + i sin (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 2 π}{3}))

簡素化

3405 (cos (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 0 π}{3}) + i sin (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 0 π}{3})) : 25 cos (\frac{arctan ( \frac{11}{2} )}{3}) + 25 i sin (\frac{arctan ( \frac{11}{2} )}{3})

簡素化

3405 (cos (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 1 π}{3}) + i sin (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 1 π}{3})) : 25 cos (\frac{arctan ( \frac{11}{2} ) + 2 π}{3}) + 25 i sin (\frac{arctan ( \frac{11}{2} ) + 2 π}{3})

簡素化

3405 (cos (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 2 π}{3}) + i sin (\frac{arctan ( \frac{11}{2} ) + 2 \cdot 2 π}{3})) : 25 cos (\frac{arctan ( \frac{11}{2} ) + 4 π}{3}) + 25 i sin (\frac{arctan ( \frac{11}{2} ) + 4 π}{3})

z = 25 cos (\frac{arctan ( \frac{11}{2} )}{3}) + 25 i sin (\frac{arctan ( \frac{11}{2} )}{3}), z = 25 cos (\frac{arctan ( \frac{11}{2} ) + 2 π}{3}) + 25 i sin (\frac{arctan ( \frac{11}{2} ) + 2 π}{3}), z = 25 cos (\frac{arctan ( \frac{11}{2} ) + 4 π}{3}) + 25 i sin (\frac{arctan ( \frac{11}{2} ) + 4 π}{3})