Trigonometry Examples

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

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 specified 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 $θ$ .

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Step 17.1

Simplify.

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Step 17.1.1

Multiply $2 π (0)$ .

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Step 17.1.1.1

Multiply $0$ by $2$ .

$3 θ = \frac{π}{2} + 0 π$

Step 17.1.1.2

Multiply $0$ by $π$ .

$3 θ = \frac{π}{2} + 0$

Step 17.1.2

Add $\frac{π}{2}$ and $0$ .

$3 θ = \frac{π}{2}$

Step 17.2

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

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Step 17.2.1

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

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

Step 17.2.2

Simplify the left side.

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Step 17.2.2.1

Cancel the common factor of $3$ .

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Step 17.2.2.1.1

Cancel the common factor.

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

Step 17.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{π}{2}}{3}$

Step 17.2.3

Simplify the right side.

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Step 17.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{π}{2} \cdot \frac{1}{3}$

Step 17.2.3.2

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

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Step 17.2.3.2.1

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

$θ = \frac{π}{2 \cdot 3}$

Step 17.2.3.2.2

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.

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Step 19.1

Simplify each term.

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Step 19.1.1

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

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

Step 19.1.2

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

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

Step 19.1.3

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

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

Step 19.2

Apply the distributive property.

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

Step 19.3

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

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Step 19.3.1

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

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

Step 19.3.2

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

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

Step 19.4

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

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

Step 19.5

Simplify each term.

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Step 19.5.1

Divide $2.18224727$ by $2$ .

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

Step 19.5.2

Factor $1.25992104$ out of $1.25992104 i$ .

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

Step 19.5.3

Factor $2$ out of $2$ .

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

Step 19.5.4

Separate fractions.

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

Step 19.5.5

Divide $1.25992104$ by $2$ .

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

Step 19.5.6

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 right 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 $θ$ .

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Step 22.1

Simplify.

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Step 22.1.1

Multiply $2$ by $1$ .

$3 θ = \frac{π}{2} + 2 π$

Step 22.1.2

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

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

Step 22.1.3

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 22.1.4

Combine the numerators over the common denominator.

$3 θ = \frac{π + 2 π \cdot 2}{2}$

Step 22.1.5

Multiply $2$ by $2$ .

$3 θ = \frac{π + 4 π}{2}$

Step 22.1.6

Add $π$ and $4 π$ .

$3 θ = \frac{5 π}{2}$

Step 22.2

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

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Step 22.2.1

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

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

Step 22.2.2

Simplify the left side.

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Step 22.2.2.1

Cancel the common factor of $3$ .

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Step 22.2.2.1.1

Cancel the common factor.

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

Step 22.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{5 π}{2}}{3}$

Step 22.2.3

Simplify the right side.

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Step 22.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{5 π}{2} \cdot \frac{1}{3}$

Step 22.2.3.2

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

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Step 22.2.3.2.1

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

$θ = \frac{5 π}{2 \cdot 3}$

Step 22.2.3.2.2

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.

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Step 24.1

Simplify each term.

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Step 24.1.1

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.

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

Step 24.1.2

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

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

Step 24.1.3

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

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

Step 24.1.4

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

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

Step 24.1.5

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

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

Step 24.2

Apply the distributive property.

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

Step 24.3

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

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Step 24.3.1

Multiply $- 1$ by $1.25992104$ .

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

Step 24.3.2

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

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

Step 24.3.3

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

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

Step 24.4

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

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

Step 24.5

Simplify each term.

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Step 24.5.1

Divide $- 2.18224727$ by $2$ .

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

Step 24.5.2

Factor $1.25992104$ out of $1.25992104 i$ .

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

Step 24.5.3

Factor $2$ out of $2$ .

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

Step 24.5.4

Separate fractions.

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

Step 24.5.5

Divide $1.25992104$ by $2$ .

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

Step 24.5.6

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 right 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 $θ$ .

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Step 27.1

Simplify.

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Step 27.1.1

Multiply $2$ by $2$ .

$3 θ = \frac{π}{2} + 4 π$

Step 27.1.2

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

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

Step 27.1.3

Combine $4 π$ and $\frac{2}{2}$ .

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

Step 27.1.4

Combine the numerators over the common denominator.

$3 θ = \frac{π + 4 π \cdot 2}{2}$

Step 27.1.5

Multiply $2$ by $4$ .

$3 θ = \frac{π + 8 π}{2}$

Step 27.1.6

Add $π$ and $8 π$ .

$3 θ = \frac{9 π}{2}$

Step 27.2

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

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Step 27.2.1

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

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

Step 27.2.2

Simplify the left side.

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Step 27.2.2.1

Cancel the common factor of $3$ .

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Step 27.2.2.1.1

Cancel the common factor.

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

Step 27.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{9 π}{2}}{3}$

Step 27.2.3

Simplify the right side.

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Step 27.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{9 π}{2} \cdot \frac{1}{3}$

Step 27.2.3.2

Cancel the common factor of $3$ .

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Step 27.2.3.2.1

Factor $3$ out of $9 π$ .

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

Step 27.2.3.2.2

Cancel the common factor.

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

Step 27.2.3.2.3

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.

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Step 29.1

Simplify each term.

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Step 29.1.1

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

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

Step 29.1.2

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

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

Step 29.1.3

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.

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

Step 29.1.4

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

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

Step 29.1.5

Multiply $- 1$ by $1$ .

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

Step 29.1.6

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

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

Step 29.1.7

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

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

Step 29.2

Simplify the expression.

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Step 29.2.1

Subtract $i$ from $0$ .

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

Step 29.2.2

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 right shift.

$z_{2} = 3 - 1.25992104 i$

Step 31

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

$z_{0} = 4.09112363 + 0.62996052 i$

$z_{1} = 1.90887636 + 0.62996052 i$

$z_{2} = 3 - 1.25992104 i$

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