# Algebra Examples

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 .

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

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 .

For

Multiply by .

For

Add and .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Multiply .

Multiply and .

Multiply by .

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

Simplify each term.

The exact value of is .

The exact value of is .

Combine and .

Simplify terms.

Apply the distributive property.

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Multiply by .

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

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

Find the value of for .

Multiply by .

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

For

Multiply by .

For

Combine the numerators over the common denominator.

Multiply by .

Add and .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Multiply .

Multiply and .

Multiply by .

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

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 .

Simplify terms.

Apply the distributive property.

Cancel the common factor of .

Move the leading negative in into the numerator.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Multiply by .

Multiply by .

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

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

Find the value of for .

Multiply by .

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

For

Multiply by .

For

Combine the numerators over the common denominator.

Multiply by .

Add and .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

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

Simplify each term.

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

The exact value of is .

Multiply by .

Move to the left of .

Rewrite as .

Simplify the expression.

Subtract from .

Multiply by .

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

These are the complex solutions to .