Answer to Question #181010 in Linear Algebra for Caylin Adams

Question #181010

Find all the third roots of i and plot them on a Argand diagram. (please also state which theorems was used) .


1
Expert's answer
2021-04-25T07:41:33-0400

Given complex number -

z=i

Real(z)=x=0Img(z)=y=1Real(z)=x=0\\ Img(z)=y=1


ϕ=arg(z)=tan1(10)=tan1()=π2\phi=arg(z)=tan^{-1}(\dfrac{1}{0})=tan^{-1}(\infty)=\dfrac{\pi}{2}


The above complex number in polar form is-


z=cosπ2+isinπ2=cos\dfrac{\pi}{2}+isin\dfrac{\pi}{2}


To find its cube root we take-


zk=z3=z3(cosϕ+2πk3+isinϕ+2πk3),k=0,1,2.z_k=\sqrt[3]{z}=\sqrt[3]{|z|}(cos {\frac {\phi+2\pi k} 3}+isin{\frac {\phi+2\pi k} 3}), k=0,1,2.


So, let's find all zkz_k


z0=(cosπ6+isinπ6)z_0=(cos {\frac {\pi} 6}+isin{\frac {\pi} 6})


z1=(cos5π6+isin5π6)z_1=(cos {\frac {5\pi} 6}+isin{\frac {5\pi} 6})


z2=(cos3π2+isin3π2)z_2=(cos {\frac {3\pi} 2}+isin{\frac {3\pi} 2})


Here Demovier's theorem is used.


Argand diagram is-



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!

Leave a comment