La calculatrice trouvera les racines n n n
Solution La forme polaire de 81 i 81 i 81 i 81 ( cos ( π 2 ) + i sin ( π 2 ) ) 81 \left(\cos{\left(\frac{\pi}{2} \right)} + i \sin{\left(\frac{\pi}{2} \right)}\right) 81 ( cos ( 2 π ) + i sin ( 2 π ) ) calculateur de forme polaire ).
Selon la formule de De Moivre, toutes les racines n n n r ( cos ( θ ) + i sin ( θ ) ) r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right) r ( cos ( θ ) + i sin ( θ ) ) r 1 n ( cos ( θ + 2 π k n ) + i sin ( θ + 2 π k n ) ) r^{\frac{1}{n}} \left(\cos{\left(\frac{\theta + 2 \pi k}{n} \right)} + i \sin{\left(\frac{\theta + 2 \pi k}{n} \right)}\right) r n 1 ( cos ( n θ + 2 πk ) + i sin ( n θ + 2 πk ) ) k = 0.. n − 1 ‾ k=\overline{0..n-1} k = 0.. n − 1
Nous avons que r = 81 r = 81 r = 81 θ = π 2 \theta = \frac{\pi}{2} θ = 2 π n = 4 n = 4 n = 4
k = 0 k = 0 k = 0 81 4 ( cos ( π 2 + 2 ⋅ π ⋅ 0 4 ) + i sin ( π 2 + 2 ⋅ π ⋅ 0 4 ) ) = 3 ( cos ( π 8 ) + i sin ( π 8 ) ) = 3 2 4 + 1 2 + 3 i 1 2 − 2 4 \sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 0}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 0}{4} \right)}\right) = 3 \left(\cos{\left(\frac{\pi}{8} \right)} + i \sin{\left(\frac{\pi}{8} \right)}\right) = 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} 4 81 ( cos ( 4 2 π + 2 ⋅ π ⋅ 0 ) + i sin ( 4 2 π + 2 ⋅ π ⋅ 0 ) ) = 3 ( cos ( 8 π ) + i sin ( 8 π ) ) = 3 4 2 + 2 1 + 3 i 2 1 − 4 2 k = 1 k = 1 k = 1 81 4 ( cos ( π 2 + 2 ⋅ π ⋅ 1 4 ) + i sin ( π 2 + 2 ⋅ π ⋅ 1 4 ) ) = 3 ( cos ( 5 π 8 ) + i sin ( 5 π 8 ) ) = − 3 1 2 − 2 4 + 3 i 2 4 + 1 2 \sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 1}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 1}{4} \right)}\right) = 3 \left(\cos{\left(\frac{5 \pi}{8} \right)} + i \sin{\left(\frac{5 \pi}{8} \right)}\right) = - 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} 4 81 ( cos ( 4 2 π + 2 ⋅ π ⋅ 1 ) + i sin ( 4 2 π + 2 ⋅ π ⋅ 1 ) ) = 3 ( cos ( 8 5 π ) + i sin ( 8 5 π ) ) = − 3 2 1 − 4 2 + 3 i 4 2 + 2 1 k = 2 k = 2 k = 2 81 4 ( cos ( π 2 + 2 ⋅ π ⋅ 2 4 ) + i sin ( π 2 + 2 ⋅ π ⋅ 2 4 ) ) = 3 ( cos ( 9 π 8 ) + i sin ( 9 π 8 ) ) = − 3 2 4 + 1 2 − 3 i 1 2 − 2 4 \sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 2}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 2}{4} \right)}\right) = 3 \left(\cos{\left(\frac{9 \pi}{8} \right)} + i \sin{\left(\frac{9 \pi}{8} \right)}\right) = - 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} 4 81 ( cos ( 4 2 π + 2 ⋅ π ⋅ 2 ) + i sin ( 4 2 π + 2 ⋅ π ⋅ 2 ) ) = 3 ( cos ( 8 9 π ) + i sin ( 8 9 π ) ) = − 3 4 2 + 2 1 − 3 i 2 1 − 4 2 k = 3 k = 3 k = 3 81 4 ( cos ( π 2 + 2 ⋅ π ⋅ 3 4 ) + i sin ( π 2 + 2 ⋅ π ⋅ 3 4 ) ) = 3 ( cos ( 13 π 8 ) + i sin ( 13 π 8 ) ) = 3 1 2 − 2 4 − 3 i 2 4 + 1 2 \sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 3}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 3}{4} \right)}\right) = 3 \left(\cos{\left(\frac{13 \pi}{8} \right)} + i \sin{\left(\frac{13 \pi}{8} \right)}\right) = 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} 4 81 ( cos ( 4 2 π + 2 ⋅ π ⋅ 3 ) + i sin ( 4 2 π + 2 ⋅ π ⋅ 3 ) ) = 3 ( cos ( 8 13 π ) + i sin ( 8 13 π ) ) = 3 2 1 − 4 2 − 3 i 4 2 + 2 1
Réponse 81 i 4 = 3 2 4 + 1 2 + 3 i 1 2 − 2 4 ≈ 2.77163859753386 + 1.148050297095269 i \sqrt[4]{81 i} = 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\approx 2.77163859753386 + 1.148050297095269 i 4 81 i = 3 4 2 + 2 1 + 3 i 2 1 − 4 2 ≈ 2.77163859753386 + 1.148050297095269 i A
81 i 4 = − 3 1 2 − 2 4 + 3 i 2 4 + 1 2 ≈ − 1.148050297095269 + 2.77163859753386 i \sqrt[4]{81 i} = - 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\approx -1.148050297095269 + 2.77163859753386 i 4 81 i = − 3 2 1 − 4 2 + 3 i 4 2 + 2 1 ≈ − 1.148050297095269 + 2.77163859753386 i A
81 i 4 = − 3 2 4 + 1 2 − 3 i 1 2 − 2 4 ≈ − 2.77163859753386 − 1.148050297095269 i \sqrt[4]{81 i} = - 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\approx -2.77163859753386 - 1.148050297095269 i 4 81 i = − 3 4 2 + 2 1 − 3 i 2 1 − 4 2 ≈ − 2.77163859753386 − 1.148050297095269 i A
81 i 4 = 3 1 2 − 2 4 − 3 i 2 4 + 1 2 ≈ 1.148050297095269 − 2.77163859753386 i \sqrt[4]{81 i} = 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\approx 1.148050297095269 - 2.77163859753386 i 4 81 i = 3 2 1 − 4 2 − 3 i 4 2 + 2 1 ≈ 1.148050297095269 − 2.77163859753386 i A
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