Find All Complex Number Solutions (z+3)^3=2i

${(z + 3)}^{3} = 2 i$

Step 1

Substitute $u$ for $z + 3$ .

$u^{3} = 2 i$

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 $b = 2$ .

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

Step 5

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

$| z | = 2$

Step 6

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

$θ = \arctan (\frac{2}{0})$

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 $| z | = 2$ .

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

Step 9

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

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

Step 10

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

$r^{3} (c o s (3 θ) + i s i n (3 θ)) = 2 i = 2 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

Step 11

Equate the modulus of the trigonometric form to $r^{3}$ to find the value of $r$ .

$r^{3} = 2$

Step 12

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

$r = \sqrt[3]{2}$

Step 13

Find the approximate value of $r$ .

$r = 1.25992104$

Step 14

Find the possible values of $θ$ .

$c o s (3 θ) = c o s (\frac{π}{2} + 2 π n)$ and $s i n (3 θ) = s i n (\frac{π}{2} + 2 π n)$

Step 15

Finding all the possible values of $θ$ leads to the equation $3 θ = \frac{π}{2} + 2 π n$ .

$3 θ = \frac{π}{2} + 2 π n$

Step 16

Find the value of $θ$ for $r = 0$ .

$3 θ = \frac{π}{2} + 2 π (0)$

Step 17

Solve the equation for $θ$ .

Simplify.

Multiply $2 π (0)$ .

Multiply $0$ by $2$ .

$3 θ = \frac{π}{2} + 0 π$

Multiply $0$ by $π$ .

$3 θ = \frac{π}{2} + 0$

Add $\frac{π}{2}$ and $0$ .

$3 θ = \frac{π}{2}$

Divide each term in $3 θ = \frac{π}{2}$ by $3$ and simplify.

Divide each term in $3 θ = \frac{π}{2}$ by $3$ .

$\frac{3 θ}{3} = \frac{\frac{π}{2}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 θ}{3} = \frac{\frac{π}{2}}{3}$

Divide $θ$ by $1$ .

$θ = \frac{\frac{π}{2}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{π}{2} \cdot \frac{1}{3}$

Multiply $\frac{π}{2} \cdot \frac{1}{3}$ .

Multiply $\frac{π}{2}$ by $\frac{1}{3}$ .

$θ = \frac{π}{2 \cdot 3}$

Multiply $2$ by $3$ .

$θ = \frac{π}{6}$

Step 18

Use the values of $θ$ and $r$ to find a solution to the equation $u^{3} = 2 i$ .

$u_{0} = 1.25992104 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6}))$

Step 19

Convert the solution to rectangular form.

Simplify each term.

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$u_{0} = 1.25992104 (\frac{\sqrt{3}}{2} + i \sin (\frac{π}{6}))$

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$u_{0} = 1.25992104 (\frac{\sqrt{3}}{2} + i (\frac{1}{2}))$

Combine $i$ and $\frac{1}{2}$ .

$u_{0} = 1.25992104 (\frac{\sqrt{3}}{2} + \frac{i}{2})$

Apply the distributive property.

$u_{0} = 1.25992104 (\frac{\sqrt{3}}{2}) + 1.25992104 (\frac{i}{2})$

Multiply $1.25992104 \frac{\sqrt{3}}{2}$ .

Combine $1.25992104$ and $\frac{\sqrt{3}}{2}$ .

$u_{0} = \frac{1.25992104 \sqrt{3}}{2} + 1.25992104 (\frac{i}{2})$

Multiply $1.25992104$ by $\sqrt{3}$ .

$u_{0} = \frac{2.18224727}{2} + 1.25992104 (\frac{i}{2})$

Combine $1.25992104$ and $\frac{i}{2}$ .

$u_{0} = \frac{2.18224727}{2} + \frac{1.25992104 i}{2}$

Simplify each term.

Divide $2.18224727$ by $2$ .

$u_{0} = 1.09112363 + \frac{1.25992104 i}{2}$

Factor $1.25992104$ out of $1.25992104 i$ .

$u_{0} = 1.09112363 + \frac{1.25992104 (i)}{2}$

Factor $2$ out of $2$ .

$u_{0} = 1.09112363 + \frac{1.25992104 (i)}{2 (1)}$

Separate fractions.

$u_{0} = 1.09112363 + \frac{1.25992104}{2} \cdot \frac{i}{1}$

Divide $1.25992104$ by $2$ .

$u_{0} = 1.09112363 + 0.62996052 (\frac{i}{1})$

Divide $i$ by $1$ .

$u_{0} = 1.09112363 + 0.62996052 i$

Step 20

Substitute $z + 3$ for $u$ to calculate the value of $z$ after the left shift.

$z_{0} = - 3 + 1.09112363 + 0.62996052 i$

Step 21

Find the value of $θ$ for $r = 1$ .

$3 θ = \frac{π}{2} + 2 π (1)$

Step 22

Solve the equation for $θ$ .

Simplify.

Multiply $2$ by $1$ .

$3 θ = \frac{π}{2} + 2 π$

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 θ = \frac{π}{2} + 2 π \cdot \frac{2}{2}$

Combine $2 π$ and $\frac{2}{2}$ .

$3 θ = \frac{π}{2} + \frac{2 π \cdot 2}{2}$

Combine the numerators over the common denominator.

$3 θ = \frac{π + 2 π \cdot 2}{2}$

Multiply $2$ by $2$ .

$3 θ = \frac{π + 4 π}{2}$

Add $π$ and $4 π$ .

$3 θ = \frac{5 π}{2}$

Divide each term in $3 θ = \frac{5 π}{2}$ by $3$ and simplify.

Divide each term in $3 θ = \frac{5 π}{2}$ by $3$ .

$\frac{3 θ}{3} = \frac{\frac{5 π}{2}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 θ}{3} = \frac{\frac{5 π}{2}}{3}$

Divide $θ$ by $1$ .

$θ = \frac{\frac{5 π}{2}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{5 π}{2} \cdot \frac{1}{3}$

Multiply $\frac{5 π}{2} \cdot \frac{1}{3}$ .

Multiply $\frac{5 π}{2}$ by $\frac{1}{3}$ .

$θ = \frac{5 π}{2 \cdot 3}$

Multiply $2$ by $3$ .

$θ = \frac{5 π}{6}$

Step 23

Use the values of $θ$ and $r$ to find a solution to the equation $u^{3} = 2 i$ .

$u_{1} = 1.25992104 (\cos (\frac{5 π}{6}) + i \sin (\frac{5 π}{6}))$

Step 24

Convert the solution to rectangular form.

Simplify each term.

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.

$u_{1} = 1.25992104 (- \cos (\frac{π}{6}) + i \sin (\frac{5 π}{6}))$

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$u_{1} = 1.25992104 (- \frac{\sqrt{3}}{2} + i \sin (\frac{5 π}{6}))$

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

$u_{1} = 1.25992104 (- \frac{\sqrt{3}}{2} + i \sin (\frac{π}{6}))$

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$u_{1} = 1.25992104 (- \frac{\sqrt{3}}{2} + i (\frac{1}{2}))$

Combine $i$ and $\frac{1}{2}$ .

$u_{1} = 1.25992104 (- \frac{\sqrt{3}}{2} + \frac{i}{2})$

Apply the distributive property.

$u_{1} = 1.25992104 (- \frac{\sqrt{3}}{2}) + 1.25992104 (\frac{i}{2})$

Multiply $1.25992104 (- \frac{\sqrt{3}}{2})$ .

Multiply $- 1$ by $1.25992104$ .

$u_{1} = - 1.25992104 \frac{\sqrt{3}}{2} + 1.25992104 (\frac{i}{2})$

Combine $- 1.25992104$ and $\frac{\sqrt{3}}{2}$ .

$u_{1} = \frac{- 1.25992104 \sqrt{3}}{2} + 1.25992104 (\frac{i}{2})$

Multiply $- 1.25992104$ by $\sqrt{3}$ .

$u_{1} = \frac{- 2.18224727}{2} + 1.25992104 (\frac{i}{2})$

Combine $1.25992104$ and $\frac{i}{2}$ .

$u_{1} = \frac{- 2.18224727}{2} + \frac{1.25992104 i}{2}$

Simplify each term.

Divide $- 2.18224727$ by $2$ .

$u_{1} = - 1.09112363 + \frac{1.25992104 i}{2}$

Factor $1.25992104$ out of $1.25992104 i$ .

$u_{1} = - 1.09112363 + \frac{1.25992104 (i)}{2}$

Factor $2$ out of $2$ .

$u_{1} = - 1.09112363 + \frac{1.25992104 (i)}{2 (1)}$

Separate fractions.

$u_{1} = - 1.09112363 + \frac{1.25992104}{2} \cdot \frac{i}{1}$

Divide $1.25992104$ by $2$ .

$u_{1} = - 1.09112363 + 0.62996052 (\frac{i}{1})$

Divide $i$ by $1$ .

$u_{1} = - 1.09112363 + 0.62996052 i$

Step 25

Substitute $z + 3$ for $u$ to calculate the value of $z$ after the left shift.

$z_{1} = - 3 - 1.09112363 + 0.62996052 i$

Step 26

Find the value of $θ$ for $r = 2$ .

$3 θ = \frac{π}{2} + 2 π (2)$

Step 27

Solve the equation for $θ$ .

Simplify.

Multiply $2$ by $2$ .

$3 θ = \frac{π}{2} + 4 π$

To write $4 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 θ = \frac{π}{2} + 4 π \cdot \frac{2}{2}$

Combine $4 π$ and $\frac{2}{2}$ .

$3 θ = \frac{π}{2} + \frac{4 π \cdot 2}{2}$

Combine the numerators over the common denominator.

$3 θ = \frac{π + 4 π \cdot 2}{2}$

Multiply $2$ by $4$ .

$3 θ = \frac{π + 8 π}{2}$

Add $π$ and $8 π$ .

$3 θ = \frac{9 π}{2}$

Divide each term in $3 θ = \frac{9 π}{2}$ by $3$ and simplify.

Divide each term in $3 θ = \frac{9 π}{2}$ by $3$ .

$\frac{3 θ}{3} = \frac{\frac{9 π}{2}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 θ}{3} = \frac{\frac{9 π}{2}}{3}$

Divide $θ$ by $1$ .

$θ = \frac{\frac{9 π}{2}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{9 π}{2} \cdot \frac{1}{3}$

Cancel the common factor of $3$ .

Factor $3$ out of $9 π$ .

$θ = \frac{3 (3 π)}{2} \cdot \frac{1}{3}$

Cancel the common factor.

$θ = \frac{3 (3 π)}{2} \cdot \frac{1}{3}$

Rewrite the expression.

$θ = \frac{3 π}{2}$

Step 28

Use the values of $θ$ and $r$ to find a solution to the equation $u^{3} = 2 i$ .

$u_{2} = 1.25992104 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$

Step 29

Convert the solution to rectangular form.

Simplify each term.

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

$u_{2} = 1.25992104 (\cos (\frac{π}{2}) + i \sin (\frac{3 π}{2}))$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$u_{2} = 1.25992104 (0 + i \sin (\frac{3 π}{2}))$

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.

$u_{2} = 1.25992104 (0 + i (- \sin (\frac{π}{2})))$

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

$u_{2} = 1.25992104 (0 + i (- 1 \cdot 1))$

Multiply $- 1$ by $1$ .

$u_{2} = 1.25992104 (0 + i \cdot - 1)$

Move $- 1$ to the left of $i$ .

$u_{2} = 1.25992104 (0 - 1 \cdot i)$

Rewrite $- 1 i$ as $- i$ .

$u_{2} = 1.25992104 (0 - i)$

Simplify the expression.

Subtract $i$ from $0$ .

$u_{2} = 1.25992104 (- i)$

Multiply $- 1$ by $1.25992104$ .

$u_{2} = - 1.25992104 i$

Step 30

Substitute $z + 3$ for $u$ to calculate the value of $z$ after the left shift.

$z_{2} = - 3 - 1.25992104 i$

Step 31

These are the complex solutions to $u^{3} = 2 i$ .

$z_{0} = - 1.90887636 + 0.62996052 i$

$z_{1} = - 4.09112363 + 0.62996052 i$

$z_{2} = - 3 - 1.25992104 i$

Find All Complex Number Solutions (z+3)^3=2i

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