Question

In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.

59. The complex fifth roots of$i$.

Short answer

The complex fifth roots of$i$are$(\cos 18+i\sin 18),(\cos 90+i\sin 90),(\cos 162+i\sin 162),(\cos 234+i\sin 234),(\cos 306+i\sin 306)$.

Step-by-step solution

Step 1. Rewrite i in the polar form.

The magnitude of $i=0+i$is

$$\begin{gathered} \sqrt{0^2+1^2}=\sqrt{1} \\ =1 \end{gathered}$$

which gives

$$\begin{gathered} i=1(0+i) \\ =0+i \\ =\cos 90°+i\sin 90° \end{gathered}$$

Step 2. Using the theorem of complex roots.

We get

$$\begin{gathered} z_k=\left[\cos \right(\frac{90}{5}+\frac{360k}{5})+\sin (\frac{90}{5}+\frac{360k}{5}\left)i\right];k=0,1,2,3,4 \\ =\left[\cos \right(18°+72°k)+i\sin (18°+72°k\left)\right] \end{gathered}$$

Step 3. Substitute k=0,1,2,3,4 in zk=[cos(18+72k)+isin(18+72k)].

We get

$$\begin{gathered} z_0=\cos 18+i\sin 18 \\ {z}_1=\cos (18+72)+i\sin (18+72) \\ =\cos 90+i\sin 90 \\ {z}_2=\cos (18+144)+i\sin (18+144) \\ =\cos 162+i\sin 162 \\ {z}_3=\cos (18+216)+i\sin (18+216) \\ =\cos 234+i\sin 234 \\ {z}_4=\cos (18+288)+i\sin (18+288)] \\ =\cos 306+i\sin 306 \end{gathered}$$

So we have found the complex fifth roots of$i$.