Precalculus Examples

$z^{6} = i$

Step 1

Substitute $u$ for $z^{6}$ .

$u^{6} = i$

Step 2

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 3

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 4

Substitute the actual values of $a = 0$ and $b = 1$ .

$| z | = \sqrt{1^{2}}$

Step 5

Pull terms out from under the radical, assuming positive real numbers.

$| z | = 1$

Step 6

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{1}{0})$

Step 7

Since the argument is undefined and $b$ is positive, the angle of the point on the complex plane is $\frac{π}{2}$ .

$θ = \frac{π}{2}$

Step 8

Substitute the values of $θ = \frac{π}{2}$ and $| z | = 1$ .

$\cos (\frac{π}{2}) + i \sin (\frac{π}{2})$

Step 9

Replace the right side of the equation with the trigonometric form.

$u^{6} = \cos (\frac{π}{2}) + i \sin (\frac{π}{2})$

Step 10

Use De Moivre's Theorem to find an equation for $u$ .

$r^{6} (c o s (6 θ) + i s i n (6 θ)) = i = \cos (\frac{π}{2}) + i \sin (\frac{π}{2})$

Step 11

Equate the modulus of the trigonometric form to $r^{6}$ to find the value of $r$ .

$r^{6} = 1$

Step 12

Solve the equation for $r$ .

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Step 12.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$r = \pm \sqrt[6]{1}$

Step 12.2

Any root of $1$ is $1$ .

$r = \pm 1$

Step 12.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 12.3.1

First, use the positive value of the $\pm$ to find the first solution.

$r = 1$

Step 12.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$r = - 1$

Step 12.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$r = 1, - 1$

Step 13

Find the approximate value of $r$ .

$r = 1$

Step 14

Find the possible values of $θ$ .

$c o s (6 θ) = c o s (2 π n)$ and $s i n (6 θ) = s i n (2 π n)$

Step 15

Finding all the possible values of $θ$ leads to the equation $6 θ = 2 π n$ .

$6 θ = 2 π n$

Step 16

Find the value of $θ$ for $r = 0$ .

$6 θ = 2 π (0)$

Step 17

Solve the equation for $θ$ .

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Step 17.1

Multiply $2 π \cdot 0$ .

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Step 17.1.1

Multiply $0$ by $2$ .

$6 θ = 0 π$

Step 17.1.2

Multiply $0$ by $π$ .

$6 θ = 0$

Step 17.2

Divide each term in $6 θ = 0$ by $6$ and simplify.

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Step 17.2.1

Divide each term in $6 θ = 0$ by $6$ .

$\frac{6 θ}{6} = \frac{0}{6}$

Step 17.2.2

Simplify the left side.

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Step 17.2.2.1

Cancel the common factor of $6$ .

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Step 17.2.2.1.1

Cancel the common factor.

$\frac{6 θ}{6} = \frac{0}{6}$

Step 17.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{0}{6}$

Step 17.2.3

Simplify the right side.

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Step 17.2.3.1

Divide $0$ by $6$ .

$θ = 0$

Step 18

Use the values of $θ$ and $r$ to find a solution to the equation $u^{6} = i$ .

$u_{0} = 1 (\cos (0) + i \sin (0))$

Step 19

Convert the solution to rectangular form.

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Step 19.1

Multiply $\cos (0) + i \sin (0)$ by $1$ .

$u_{0} = \cos (0) + i \sin (0)$

Step 19.2

Simplify each term.

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Step 19.2.1

The exact value of $\cos (0)$ is $1$ .

$u_{0} = 1 + i \sin (0)$

Step 19.2.2

The exact value of $\sin (0)$ is $0$ .

$u_{0} = 1 + i \cdot 0$

Step 19.2.3

Multiply $i$ by $0$ .

$u_{0} = 1 + 0$

Step 19.3

Add $1$ and $0$ .

$u_{0} = 1$

Step 20

Substitute $z^{6}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{0} = 0 + 1$

Step 21

Find the value of $θ$ for $r = 1$ .

$6 θ = 2 π (1)$

Step 22

Solve the equation for $θ$ .

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Step 22.1

Multiply $2$ by $1$ .

$6 θ = 2 π$

Step 22.2

Divide each term in $6 θ = 2 π$ by $6$ and simplify.

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Step 22.2.1

Divide each term in $6 θ = 2 π$ by $6$ .

$\frac{6 θ}{6} = \frac{2 π}{6}$

Step 22.2.2

Simplify the left side.

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Step 22.2.2.1

Cancel the common factor of $6$ .

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Step 22.2.2.1.1

Cancel the common factor.

$\frac{6 θ}{6} = \frac{2 π}{6}$

Step 22.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{2 π}{6}$

Step 22.2.3

Simplify the right side.

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Step 22.2.3.1

Cancel the common factor of $2$ and $6$ .

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Step 22.2.3.1.1

Factor $2$ out of $2 π$ .

$θ = \frac{2 (π)}{6}$

Step 22.2.3.1.2

Cancel the common factors.

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Step 22.2.3.1.2.1

Factor $2$ out of $6$ .

$θ = \frac{2 π}{2 \cdot 3}$

Step 22.2.3.1.2.2

Cancel the common factor.

$θ = \frac{2 π}{2 \cdot 3}$

Step 22.2.3.1.2.3

Rewrite the expression.

$θ = \frac{π}{3}$

Step 23

Use the values of $θ$ and $r$ to find a solution to the equation $u^{6} = i$ .

$u_{1} = 1 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3}))$

Step 24

Convert the solution to rectangular form.

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Step 24.1

Multiply $\cos (\frac{π}{3}) + i \sin (\frac{π}{3})$ by $1$ .

$u_{1} = \cos (\frac{π}{3}) + i \sin (\frac{π}{3})$

Step 24.2

Simplify each term.

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Step 24.2.1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$u_{1} = \frac{1}{2} + i \sin (\frac{π}{3})$

Step 24.2.2

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$u_{1} = \frac{1}{2} + i (\frac{\sqrt{3}}{2})$

Step 24.2.3

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

$u_{1} = \frac{1}{2} + \frac{i \sqrt{3}}{2}$

Step 25

Substitute $z^{6}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{1} = 0 + \frac{1}{2} + \frac{i \sqrt{3}}{2}$

Step 26

Find the value of $θ$ for $r = 2$ .

$6 θ = 2 π (2)$

Step 27

Solve the equation for $θ$ .

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Step 27.1

Multiply $2$ by $2$ .

$6 θ = 4 π$

Step 27.2

Divide each term in $6 θ = 4 π$ by $6$ and simplify.

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Step 27.2.1

Divide each term in $6 θ = 4 π$ by $6$ .

$\frac{6 θ}{6} = \frac{4 π}{6}$

Step 27.2.2

Simplify the left side.

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Step 27.2.2.1

Cancel the common factor of $6$ .

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Step 27.2.2.1.1

Cancel the common factor.

$\frac{6 θ}{6} = \frac{4 π}{6}$

Step 27.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{4 π}{6}$

Step 27.2.3

Simplify the right side.

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Step 27.2.3.1

Cancel the common factor of $4$ and $6$ .

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Step 27.2.3.1.1

Factor $2$ out of $4 π$ .

$θ = \frac{2 (2 π)}{6}$

Step 27.2.3.1.2

Cancel the common factors.

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Step 27.2.3.1.2.1

Factor $2$ out of $6$ .

$θ = \frac{2 (2 π)}{2 (3)}$

Step 27.2.3.1.2.2

Cancel the common factor.

$θ = \frac{2 (2 π)}{2 \cdot 3}$

Step 27.2.3.1.2.3

Rewrite the expression.

$θ = \frac{2 π}{3}$

Step 28

Use the values of $θ$ and $r$ to find a solution to the equation $u^{6} = i$ .

$u_{2} = 1 (\cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3}))$

Step 29

Convert the solution to rectangular form.

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Step 29.1

Multiply $\cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3})$ by $1$ .

$u_{2} = \cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3})$

Step 29.2

Simplify each term.

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Step 29.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$u_{2} = - \cos (\frac{π}{3}) + i \sin (\frac{2 π}{3})$

Step 29.2.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$u_{2} = - \frac{1}{2} + i \sin (\frac{2 π}{3})$

Step 29.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$u_{2} = - \frac{1}{2} + i \sin (\frac{π}{3})$

Step 29.2.4

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$u_{2} = - \frac{1}{2} + i (\frac{\sqrt{3}}{2})$

Step 29.2.5

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

$u_{2} = - \frac{1}{2} + \frac{i \sqrt{3}}{2}$

Step 30

Substitute $z^{6}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{2} = 0 - \frac{1}{2} + \frac{i \sqrt{3}}{2}$

Step 31

Find the value of $θ$ for $r = 3$ .

$6 θ = 2 π (3)$

Step 32

Solve the equation for $θ$ .

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Step 32.1

Multiply $3$ by $2$ .

$6 θ = 6 π$

Step 32.2

Divide each term in $6 θ = 6 π$ by $6$ and simplify.

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Step 32.2.1

Divide each term in $6 θ = 6 π$ by $6$ .

$\frac{6 θ}{6} = \frac{6 π}{6}$

Step 32.2.2

Simplify the left side.

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Step 32.2.2.1

Cancel the common factor of $6$ .

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Step 32.2.2.1.1

Cancel the common factor.

$\frac{6 θ}{6} = \frac{6 π}{6}$

Step 32.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{6 π}{6}$

Step 32.2.3

Simplify the right side.

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Step 32.2.3.1

Cancel the common factor of $6$ .

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Step 32.2.3.1.1

Cancel the common factor.

$θ = \frac{6 π}{6}$

Step 32.2.3.1.2

Divide $π$ by $1$ .

$θ = π$

Step 33

Use the values of $θ$ and $r$ to find a solution to the equation $u^{6} = i$ .

$u_{3} = 1 (\cos (π) + i \sin (π))$

Step 34

Convert the solution to rectangular form.

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Step 34.1

Multiply $\cos (π) + i \sin (π)$ by $1$ .

$u_{3} = \cos (π) + i \sin (π)$

Step 34.2

Simplify each term.

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Step 34.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$u_{3} = - \cos (0) + i \sin (π)$

Step 34.2.2

The exact value of $\cos (0)$ is $1$ .

$u_{3} = - 1 \cdot 1 + i \sin (π)$

Step 34.2.3

Multiply $- 1$ by $1$ .

$u_{3} = - 1 + i \sin (π)$

Step 34.2.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$u_{3} = - 1 + i \sin (0)$

Step 34.2.5

The exact value of $\sin (0)$ is $0$ .

$u_{3} = - 1 + i \cdot 0$

Step 34.2.6

Multiply $i$ by $0$ .

$u_{3} = - 1 + 0$

Step 34.3

Add $- 1$ and $0$ .

$u_{3} = - 1$

Step 35

Substitute $z^{6}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{3} = 0 - 1$

Step 36

Find the value of $θ$ for $r = 4$ .

$6 θ = 2 π (4)$

Step 37

Solve the equation for $θ$ .

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Step 37.1

Multiply $4$ by $2$ .

$6 θ = 8 π$

Step 37.2

Divide each term in $6 θ = 8 π$ by $6$ and simplify.

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Step 37.2.1

Divide each term in $6 θ = 8 π$ by $6$ .

$\frac{6 θ}{6} = \frac{8 π}{6}$

Step 37.2.2

Simplify the left side.

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Step 37.2.2.1

Cancel the common factor of $6$ .

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Step 37.2.2.1.1

Cancel the common factor.

$\frac{6 θ}{6} = \frac{8 π}{6}$

Step 37.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{8 π}{6}$

Step 37.2.3

Simplify the right side.

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Step 37.2.3.1

Cancel the common factor of $8$ and $6$ .

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Step 37.2.3.1.1

Factor $2$ out of $8 π$ .

$θ = \frac{2 (4 π)}{6}$

Step 37.2.3.1.2

Cancel the common factors.

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Step 37.2.3.1.2.1

Factor $2$ out of $6$ .

$θ = \frac{2 (4 π)}{2 (3)}$

Step 37.2.3.1.2.2

Cancel the common factor.

$θ = \frac{2 (4 π)}{2 \cdot 3}$

Step 37.2.3.1.2.3

Rewrite the expression.

$θ = \frac{4 π}{3}$

Step 38

Use the values of $θ$ and $r$ to find a solution to the equation $u^{6} = i$ .

$u_{4} = 1 (\cos (\frac{4 π}{3}) + i \sin (\frac{4 π}{3}))$

Step 39

Convert the solution to rectangular form.

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Step 39.1

Multiply $\cos (\frac{4 π}{3}) + i \sin (\frac{4 π}{3})$ by $1$ .

$u_{4} = \cos (\frac{4 π}{3}) + i \sin (\frac{4 π}{3})$

Step 39.2

Simplify each term.

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Step 39.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$u_{4} = - \cos (\frac{π}{3}) + i \sin (\frac{4 π}{3})$

Step 39.2.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$u_{4} = - \frac{1}{2} + i \sin (\frac{4 π}{3})$

Step 39.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$u_{4} = - \frac{1}{2} + i (- \sin (\frac{π}{3}))$

Step 39.2.4

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$u_{4} = - \frac{1}{2} + i (- \frac{\sqrt{3}}{2})$

Step 39.2.5

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

$u_{4} = - \frac{1}{2} - \frac{i \sqrt{3}}{2}$

Step 40

Substitute $z^{6}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{4} = 0 - \frac{1}{2} - \frac{i \sqrt{3}}{2}$

Step 41

Find the value of $θ$ for $r = 5$ .

$6 θ = 2 π (5)$

Step 42

Solve the equation for $θ$ .

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Step 42.1

Multiply $5$ by $2$ .

$6 θ = 10 π$

Step 42.2

Divide each term in $6 θ = 10 π$ by $6$ and simplify.

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Step 42.2.1

Divide each term in $6 θ = 10 π$ by $6$ .

$\frac{6 θ}{6} = \frac{10 π}{6}$

Step 42.2.2

Simplify the left side.

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Step 42.2.2.1

Cancel the common factor of $6$ .

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Step 42.2.2.1.1

Cancel the common factor.

$\frac{6 θ}{6} = \frac{10 π}{6}$

Step 42.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{10 π}{6}$

Step 42.2.3

Simplify the right side.

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Step 42.2.3.1

Cancel the common factor of $10$ and $6$ .

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Step 42.2.3.1.1

Factor $2$ out of $10 π$ .

$θ = \frac{2 (5 π)}{6}$

Step 42.2.3.1.2

Cancel the common factors.

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Step 42.2.3.1.2.1

Factor $2$ out of $6$ .

$θ = \frac{2 (5 π)}{2 (3)}$

Step 42.2.3.1.2.2

Cancel the common factor.

$θ = \frac{2 (5 π)}{2 \cdot 3}$

Step 42.2.3.1.2.3

Rewrite the expression.

$θ = \frac{5 π}{3}$

Step 43

Use the values of $θ$ and $r$ to find a solution to the equation $u^{6} = i$ .

$u_{5} = 1 (\cos (\frac{5 π}{3}) + i \sin (\frac{5 π}{3}))$

Step 44

Convert the solution to rectangular form.

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Step 44.1

Multiply $\cos (\frac{5 π}{3}) + i \sin (\frac{5 π}{3})$ by $1$ .

$u_{5} = \cos (\frac{5 π}{3}) + i \sin (\frac{5 π}{3})$

Step 44.2

Simplify each term.

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Step 44.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$u_{5} = \cos (\frac{π}{3}) + i \sin (\frac{5 π}{3})$

Step 44.2.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$u_{5} = \frac{1}{2} + i \sin (\frac{5 π}{3})$

Step 44.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$u_{5} = \frac{1}{2} + i (- \sin (\frac{π}{3}))$

Step 44.2.4

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$u_{5} = \frac{1}{2} + i (- \frac{\sqrt{3}}{2})$

Step 44.2.5

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

$u_{5} = \frac{1}{2} - \frac{i \sqrt{3}}{2}$

Step 45

Substitute $z^{6}$ for $u$ to calculate the value of $z$ after the right shift.

$z_{5} = 0 + \frac{1}{2} - \frac{i \sqrt{3}}{2}$

Step 46

These are the complex solutions to $u^{6} = i$ .

$z_{0} = 1$

$z_{1} = \frac{1}{2} + \frac{i \sqrt{3}}{2}$

$z_{2} = - \frac{1}{2} + \frac{i \sqrt{3}}{2}$

$z_{3} = - 1$

$z_{4} = - \frac{1}{2} - \frac{i \sqrt{3}}{2}$

$z_{5} = \frac{1}{2} - \frac{i \sqrt{3}}{2}$