Substitute for .

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

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

where

Substitute the actual values of and .

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

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

Since the argument is undefined and is positive, the angle of the point on the complex plane is .

Substitute the values of and .

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

Use De Moivre’s Theorem to find an equation for .

Equate the modulus of the trigonometric form to to find the value of .

Move to the left side of the equation by subtracting it from both sides.

Factor the left side of the equation.

Rewrite as .

Since both terms are perfect cubes, factor using the difference of cubes formula, where and .

Simplify.

Multiply by .

One to any power is one.

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Set the first factor equal to and solve.

Set the first factor equal to .

Add to both sides of the equation.

Set the next factor equal to and solve.

Set the next factor equal to .

Use the quadratic formula to find the solutions.

Substitute the values , , and into the quadratic formula and solve for .

Simplify.

Simplify the numerator.

One to any power is one.

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Multiply by .

Simplify the expression to solve for the portion of the .

Simplify the numerator.

One to any power is one.

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Multiply by .

Change the to .

Rewrite as .

Factor out of .

Factor out of .

Move the negative in front of the fraction.

Simplify the expression to solve for the portion of the .

Simplify the numerator.

One to any power is one.

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Multiply by .

Change the to .

Rewrite as .

Factor out of .

Factor out of .

Move the negative in front of the fraction.

The final answer is the combination of both solutions.

The final solution is all the values that make true.

Find the approximate value of .

Find the possible values of .

and

Finding all the possible values of leads to the equation .

Find the value of for .

Multiply .

Multiply by .

Multiply by .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Use the values of and to find a solution to the equation .

Multiply by .

Simplify each term.

The exact value of is .

The exact value of is .

Multiply by .

Add and .

Substitute for to calculate the value of after the right shift.

Find the value of for .

Multiply by .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Use the values of and to find a solution to the equation .

Multiply by .

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.

The exact value of is .

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

The exact value of is .

Combine and .

Substitute for to calculate the value of after the right shift.

Find the value of for .

Multiply by .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Use the values of and to find a solution to the equation .

Multiply by .

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 third quadrant.

The exact value of is .

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.

The exact value of is .

Combine and .

Substitute for to calculate the value of after the right shift.

These are the complex solutions to .

Find All Complex Number Solutions z^3=i