Question #142635
Using De Moivre's theorem or otherwise
find the six roots of the equation z^6 +1 = 0
giving your answers in the form e^iw where w= angle teta
1
Expert's answer
2020-11-23T19:22:12-0500

Let us find the six roots of the equation z6=1z^6=-1 using de Moivre's Theorem for fractional power:


[r(cosθ+isinθ)]1n=rn(cosθ+2πkn+isinθ+2πkn)[r(\cos\theta+i\sin\theta)]^{\frac{1}{n}}=\sqrt[n]{r}(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}) for k{0,1,...,n1}.k\in\{0,1,...,n-1\}.


In trigonometric form 1=cosπ+isinπ-1=\cos\pi+i\sin\pi, that is r=1,θ=π.r=1, \theta=\pi. Consequently, the formula


zk=cosπ+2πk6+isinπ+2πk6=eπ+2πk6iz_k=\cos\frac{\pi+2\pi k}{6}+i\sin\frac{\pi+2\pi k}{6}=e^{\frac{\pi+2\pi k}{6}i} for k{0,1,2,3,4,5}k\in\{0,1,2,3,4,5\} gives all six different roots.


Therefore, we have:


z0=eπ6iz_0=e^{\frac{\pi}{6}i}, z1=e3π6i=eπ2iz_1=e^{\frac{3\pi}{6}i}=e^{\frac{\pi}{2}i}, z2=e5π6iz_2=e^{\frac{5\pi}{6}i}, z3=e7π6iz_3=e^{\frac{7\pi}{6}i}, z4=e9π6i=e3π2iz_4=e^{\frac{9\pi}{6}i}=e^{\frac{3\pi}{2}i}, z5=e11π6i.z_5=e^{\frac{11\pi}{6}i}.




Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS