# Algebra Examples

Step 1

Substitute for .

Step 2

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

Step 3

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

where

Step 4

Substitute the actual values of and .

Step 5

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

Step 6

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

Step 7

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

Step 8

Substitute the values of and .

Step 9

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

Step 10

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

Step 11

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

Step 12

Step 12.1

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

Step 12.2

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

Step 12.2.1

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

Step 12.2.2

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

Step 12.2.3

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

Step 13

Find the approximate value of .

Step 14

Find the possible values of .

and

Step 15

Finding all the possible values of leads to the equation .

Step 16

Find the value of for .

Step 17

Step 17.1

Simplify.

Step 17.1.1

Multiply .

Step 17.1.1.1

Multiply by .

Step 17.1.1.2

Multiply by .

Step 17.1.2

Add and .

Step 17.2

Divide each term in by and simplify.

Step 17.2.1

Divide each term in by .

Step 17.2.2

Simplify the left side.

Step 17.2.2.1

Cancel the common factor of .

Step 17.2.2.1.1

Cancel the common factor.

Step 17.2.2.1.2

Divide by .

Step 17.2.3

Simplify the right side.

Step 17.2.3.1

Multiply the numerator by the reciprocal of the denominator.

Step 17.2.3.2

Multiply .

Step 17.2.3.2.1

Multiply by .

Step 17.2.3.2.2

Multiply by .

Step 18

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

Step 19

Step 19.1

Simplify each term.

Step 19.1.1

The exact value of is .

Step 19.1.1.1

Rewrite as an angle where the values of the six trigonometric functions are known divided by .

Step 19.1.1.2

Apply the cosine half-angle identity .

Step 19.1.1.3

Change the to because cosine is positive in the first quadrant.

Step 19.1.1.4

The exact value of is .

Step 19.1.1.5

Simplify .

Step 19.1.1.5.1

Write as a fraction with a common denominator.

Step 19.1.1.5.2

Combine the numerators over the common denominator.

Step 19.1.1.5.3

Multiply the numerator by the reciprocal of the denominator.

Step 19.1.1.5.4

Multiply .

Step 19.1.1.5.4.1

Multiply by .

Step 19.1.1.5.4.2

Multiply by .

Step 19.1.1.5.5

Rewrite as .

Step 19.1.1.5.6

Simplify the denominator.

Step 19.1.1.5.6.1

Rewrite as .

Step 19.1.1.5.6.2

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

Step 19.1.2

The exact value of is .

Step 19.1.2.1

Rewrite as an angle where the values of the six trigonometric functions are known divided by .

Step 19.1.2.2

Apply the sine half-angle identity.

Step 19.1.2.3

Change the to because sine is positive in the first quadrant.

Step 19.1.2.4

Simplify .

Step 19.1.2.4.1

The exact value of is .

Step 19.1.2.4.2

Write as a fraction with a common denominator.

Step 19.1.2.4.3

Combine the numerators over the common denominator.

Step 19.1.2.4.4

Multiply the numerator by the reciprocal of the denominator.

Step 19.1.2.4.5

Multiply .

Step 19.1.2.4.5.1

Multiply by .

Step 19.1.2.4.5.2

Multiply by .

Step 19.1.2.4.6

Rewrite as .

Step 19.1.2.4.7

Simplify the denominator.

Step 19.1.2.4.7.1

Rewrite as .

Step 19.1.2.4.7.2

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

Step 19.1.3

Combine and .

Step 19.2

Simplify terms.

Step 19.2.1

Combine the numerators over the common denominator.

Step 19.2.2

Combine and .

Step 19.2.3

Factor out of .

Step 19.3

Separate fractions.

Step 19.4

Simplify the expression.

Step 19.4.1

Divide by .

Step 19.4.2

Divide by .

Step 19.5

Apply the distributive property.

Step 19.6

Multiply by .

Step 19.7

Multiply by .

Step 20

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

Step 21

Find the value of for .

Step 22

Step 22.1

Simplify.

Step 22.1.1

Multiply by .

Step 22.1.2

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

Step 22.1.3

Combine and .

Step 22.1.4

Combine the numerators over the common denominator.

Step 22.1.5

Multiply by .

Step 22.1.6

Add and .

Step 22.2

Divide each term in by and simplify.

Step 22.2.1

Divide each term in by .

Step 22.2.2

Simplify the left side.

Step 22.2.2.1

Cancel the common factor of .

Step 22.2.2.1.1

Cancel the common factor.

Step 22.2.2.1.2

Divide by .

Step 22.2.3

Simplify the right side.

Step 22.2.3.1

Multiply the numerator by the reciprocal of the denominator.

Step 22.2.3.2

Multiply .

Step 22.2.3.2.1

Multiply by .

Step 22.2.3.2.2

Multiply by .

Step 23

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

Step 24

Step 24.1

Simplify each term.

Step 24.1.1

The exact value of is .

Step 24.1.1.1

Rewrite as an angle where the values of the six trigonometric functions are known divided by .

Step 24.1.1.2

Apply the cosine half-angle identity .

Step 24.1.1.3

Change the to because cosine is negative in the second quadrant.

Step 24.1.1.4

Simplify .

Step 24.1.1.4.1

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 third quadrant.

Step 24.1.1.4.2

The exact value of is .

Step 24.1.1.4.3

Write as a fraction with a common denominator.

Step 24.1.1.4.4

Combine the numerators over the common denominator.

Step 24.1.1.4.5

Multiply the numerator by the reciprocal of the denominator.

Step 24.1.1.4.6

Multiply .

Step 24.1.1.4.6.1

Multiply by .

Step 24.1.1.4.6.2

Multiply by .

Step 24.1.1.4.7

Rewrite as .

Step 24.1.1.4.8

Simplify the denominator.

Step 24.1.1.4.8.1

Rewrite as .

Step 24.1.1.4.8.2

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

Step 24.1.2

The exact value of is .

Step 24.1.2.1

Rewrite as an angle where the values of the six trigonometric functions are known divided by .

Step 24.1.2.2

Apply the sine half-angle identity.

Step 24.1.2.3

Change the to because sine is positive in the second quadrant.

Step 24.1.2.4

Simplify .

Step 24.1.2.4.1

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 third quadrant.

Step 24.1.2.4.2

The exact value of is .

Step 24.1.2.4.3

Multiply .

Step 24.1.2.4.3.1

Multiply by .

Step 24.1.2.4.3.2

Multiply by .

Step 24.1.2.4.4

Write as a fraction with a common denominator.

Step 24.1.2.4.5

Combine the numerators over the common denominator.

Step 24.1.2.4.6

Multiply the numerator by the reciprocal of the denominator.

Step 24.1.2.4.7

Multiply .

Step 24.1.2.4.7.1

Multiply by .

Step 24.1.2.4.7.2

Multiply by .

Step 24.1.2.4.8

Rewrite as .

Step 24.1.2.4.9

Simplify the denominator.

Step 24.1.2.4.9.1

Rewrite as .

Step 24.1.2.4.9.2

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

Step 24.1.3

Combine and .

Step 24.2

Simplify terms.

Step 24.2.1

Combine the numerators over the common denominator.

Step 24.2.2

Combine and .

Step 24.2.3

Factor out of .

Step 24.3

Separate fractions.

Step 24.4

Simplify the expression.

Step 24.4.1

Divide by .

Step 24.4.2

Divide by .

Step 24.5

Apply the distributive property.

Step 24.6

Multiply .

Step 24.6.1

Multiply by .

Step 24.6.2

Multiply by .

Step 24.7

Multiply by .

Step 25

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

Step 26

Find the value of for .

Step 27

Step 27.1

Simplify.

Step 27.1.1

Multiply by .

Step 27.1.2

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

Step 27.1.3

Combine and .

Step 27.1.4

Combine the numerators over the common denominator.

Step 27.1.5

Multiply by .

Step 27.1.6

Add and .

Step 27.2

Divide each term in by and simplify.

Step 27.2.1

Divide each term in by .

Step 27.2.2

Simplify the left side.

Step 27.2.2.1

Cancel the common factor of .

Step 27.2.2.1.1

Cancel the common factor.

Step 27.2.2.1.2

Divide by .

Step 27.2.3

Simplify the right side.

Step 27.2.3.1

Multiply the numerator by the reciprocal of the denominator.

Step 27.2.3.2

Multiply .

Step 27.2.3.2.1

Multiply by .

Step 27.2.3.2.2

Multiply by .

Step 28

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

Step 29

Step 29.1

Simplify each term.

Step 29.1.1

The exact value of is .

Step 29.1.1.1

Rewrite as an angle where the values of the six trigonometric functions are known divided by .

Step 29.1.1.2

Apply the cosine half-angle identity .

Step 29.1.1.3

Change the to because cosine is negative in the third quadrant.

Step 29.1.1.4

Simplify .

Step 29.1.1.4.1

Subtract full rotations of until the angle is greater than or equal to and less than .

Step 29.1.1.4.2

The exact value of is .

Step 29.1.1.4.3

Write as a fraction with a common denominator.

Step 29.1.1.4.4

Combine the numerators over the common denominator.

Step 29.1.1.4.5

Multiply the numerator by the reciprocal of the denominator.

Step 29.1.1.4.6

Multiply .

Step 29.1.1.4.6.1

Multiply by .

Step 29.1.1.4.6.2

Multiply by .

Step 29.1.1.4.7

Rewrite as .

Step 29.1.1.4.8

Simplify the denominator.

Step 29.1.1.4.8.1

Rewrite as .

Step 29.1.1.4.8.2

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

Step 29.1.2

The exact value of is .

Step 29.1.2.1

Rewrite as an angle where the values of the six trigonometric functions are known divided by .

Step 29.1.2.2

Apply the sine half-angle identity.

Step 29.1.2.3

Change the to because sine is negative in the third quadrant.

Step 29.1.2.4

Simplify .

Step 29.1.2.4.1

Subtract full rotations of until the angle is greater than or equal to and less than .

Step 29.1.2.4.2

The exact value of is .

Step 29.1.2.4.3

Write as a fraction with a common denominator.

Step 29.1.2.4.4

Combine the numerators over the common denominator.

Step 29.1.2.4.5

Multiply the numerator by the reciprocal of the denominator.

Step 29.1.2.4.6

Multiply .

Step 29.1.2.4.6.1

Multiply by .

Step 29.1.2.4.6.2

Multiply by .

Step 29.1.2.4.7

Rewrite as .

Step 29.1.2.4.8

Simplify the denominator.

Step 29.1.2.4.8.1

Rewrite as .

Step 29.1.2.4.8.2

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

Step 29.1.3

Combine and .

Step 29.2

Simplify terms.

Step 29.2.1

Combine the numerators over the common denominator.

Step 29.2.2

Combine and .

Step 29.2.3

Factor out of .

Step 29.3

Separate fractions.

Step 29.4

Simplify the expression.

Step 29.4.1

Divide by .

Step 29.4.2

Divide by .

Step 29.5

Apply the distributive property.

Step 29.6

Multiply .

Step 29.6.1

Multiply by .

Step 29.6.2

Multiply by .

Step 29.7

Multiply .

Step 29.7.1

Multiply by .

Step 29.7.2

Multiply by .

Step 30

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

Step 31

Find the value of for .

Step 32

Step 32.1

Simplify.

Step 32.1.1

Multiply by .

Step 32.1.2

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

Step 32.1.3

Combine and .

Step 32.1.4

Combine the numerators over the common denominator.

Step 32.1.5

Multiply by .

Step 32.1.6

Add and .

Step 32.2

Divide each term in by and simplify.

Step 32.2.1

Divide each term in by .

Step 32.2.2

Simplify the left side.

Step 32.2.2.1

Cancel the common factor of .

Step 32.2.2.1.1

Cancel the common factor.

Step 32.2.2.1.2

Divide by .

Step 32.2.3

Simplify the right side.

Step 32.2.3.1

Multiply the numerator by the reciprocal of the denominator.

Step 32.2.3.2

Multiply .

Step 32.2.3.2.1

Multiply by .

Step 32.2.3.2.2

Multiply by .

Step 33

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

Step 34

Step 34.1

Simplify each term.

Step 34.1.1

The exact value of is .

Step 34.1.1.1

Rewrite as an angle where the values of the six trigonometric functions are known divided by .

Step 34.1.1.2

Apply the cosine half-angle identity .

Step 34.1.1.3

Change the to because cosine is positive in the fourth quadrant.

Step 34.1.1.4

Simplify .

Step 34.1.1.4.1

Subtract full rotations of until the angle is greater than or equal to and less than .

Step 34.1.1.4.2

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.

Step 34.1.1.4.3

The exact value of is .

Step 34.1.1.4.4

Write as a fraction with a common denominator.

Step 34.1.1.4.5

Combine the numerators over the common denominator.

Step 34.1.1.4.6

Multiply the numerator by the reciprocal of the denominator.

Step 34.1.1.4.7

Multiply .

Step 34.1.1.4.7.1

Multiply by .

Step 34.1.1.4.7.2

Multiply by .

Step 34.1.1.4.8

Rewrite as .

Step 34.1.1.4.9

Simplify the denominator.

Step 34.1.1.4.9.1

Rewrite as .

Step 34.1.1.4.9.2

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

Step 34.1.2

The exact value of is .

Step 34.1.2.1

Rewrite as an angle where the values of the six trigonometric functions are known divided by .

Step 34.1.2.2

Apply the sine half-angle identity.

Step 34.1.2.3

Change the to because sine is negative in the fourth quadrant.

Step 34.1.2.4

Simplify .

Step 34.1.2.4.1

Subtract full rotations of until the angle is greater than or equal to and less than .

Step 34.1.2.4.2

Step 34.1.2.4.3

The exact value of is .

Step 34.1.2.4.4

Multiply .

Step 34.1.2.4.4.1

Multiply by .

Step 34.1.2.4.4.2

Multiply by .

Step 34.1.2.4.5

Write as a fraction with a common denominator.

Step 34.1.2.4.6

Combine the numerators over the common denominator.

Step 34.1.2.4.7

Multiply the numerator by the reciprocal of the denominator.

Step 34.1.2.4.8

Multiply .

Step 34.1.2.4.8.1

Multiply by .

Step 34.1.2.4.8.2

Multiply by .

Step 34.1.2.4.9

Rewrite as .

Step 34.1.2.4.10

Simplify the denominator.

Step 34.1.2.4.10.1

Rewrite as .

Step 34.1.2.4.10.2

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

Step 34.1.3

Combine and .

Step 34.2

Simplify terms.

Step 34.2.1

Combine the numerators over the common denominator.

Step 34.2.2

Combine and .

Step 34.2.3

Factor out of .

Step 34.3

Separate fractions.

Step 34.4

Simplify the expression.

Step 34.4.1

Divide by .

Step 34.4.2

Divide by .

Step 34.5

Apply the distributive property.

Step 34.6

Multiply by .

Step 34.7

Multiply .

Step 34.7.1

Multiply by .

Step 34.7.2

Multiply by .

Step 35

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

Step 36

These are the complex solutions to .