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

$θ$ .

Tap for more steps...

Step 17.1

Simplify.

Tap for more steps...

Step 17.1.1

Multiply

$2 π (0)$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 17.2.2.1

Cancel the common factor of

$3$ .

Tap for more steps...

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.

Tap for more steps...

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

$\frac{π}{2} \cdot \frac{1}{3}$ .

Tap for more steps...

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.

Tap for more steps...

Step 19.1

Simplify each term.

Tap for more steps...

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

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

Tap for more steps...

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.

Tap for more steps...

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 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

$θ$ .

Tap for more steps...

Step 22.1

Simplify.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 22.2.2.1

Cancel the common factor of

$3$ .

Tap for more steps...

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.

Tap for more steps...

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

$\frac{5 π}{2} \cdot \frac{1}{3}$ .

Tap for more steps...

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.

Tap for more steps...

Step 24.1

Simplify each term.

Tap for more steps...

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

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

Tap for more steps...

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.

Tap for more steps...

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 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

$θ$ .

Tap for more steps...

Step 27.1

Simplify.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 27.2.2.1

Cancel the common factor of

$3$ .

Tap for more steps...

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.

Tap for more steps...

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

$3$ .

Tap for more steps...

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.

Tap for more steps...

Step 29.1

Simplify each term.

Tap for more steps...

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.

Tap for more steps...

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 left shift.

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

Step 31

These are the complex solutions to

$u^{3} = 2 i$ .

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

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

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

Enter YOUR Problem