# Trigonometry 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 4th root of both sides of the equation to eliminate the exponent on the left side.

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the to find the first solution.

Next, use the negative value of the to find the second solution.

The complete solution is the result of both the positive and negative portions of the solution.

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 .

Reduce the expression by cancelling the common factors.

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.

Evaluate .

Evaluate .

Move to the left of .

Simplify by multiplying through.

Apply the distributive property.

Multiply.

Multiply by .

Multiply by .

Substitute for to calculate the value of after the right 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 .

Reduce the expression by cancelling the common factors.

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.

Evaluate .

Evaluate .

Move to the left of .

Simplify by multiplying through.

Apply the distributive property.

Multiply.

Multiply by .

Multiply by .

Substitute for to calculate the value of after the right 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 .

Reduce the expression by cancelling the common factors.

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.

Evaluate .

Evaluate .

Move to the left of .

Simplify by multiplying through.

Apply the distributive property.

Multiply.

Multiply by .

Multiply by .

Substitute for to calculate the value of after the right 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 .

Reduce the expression by cancelling the common factors.

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.

Evaluate .

Evaluate .

Move to the left of .

Simplify by multiplying through.

Apply the distributive property.

Multiply.

Multiply by .

Multiply by .

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

These are the complex solutions to .