# Trigonometry Examples

,

Step 1

Calculate the distance from to the origin using the formula .

Step 2

Step 2.1

Simplify the expression.

Step 2.1.1

Raise to the power of .

Step 2.1.2

Move to the left of .

Step 2.1.3

Apply the product rule to .

Step 2.1.4

Raise to the power of .

Step 2.2

Rewrite as .

Step 2.2.1

Use to rewrite as .

Step 2.2.2

Apply the power rule and multiply exponents, .

Step 2.2.3

Combine and .

Step 2.2.4

Cancel the common factor of .

Step 2.2.4.1

Cancel the common factor.

Step 2.2.4.2

Rewrite the expression.

Step 2.2.5

Evaluate the exponent.

Step 2.3

Simplify the expression.

Step 2.3.1

Multiply by .

Step 2.3.2

Add and .

Step 2.3.3

Rewrite as .

Step 2.3.4

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

Step 3

Calculate reference angle .

Step 4

Step 4.1

Cancel the common factor of .

Step 4.1.1

Cancel the common factor.

Step 4.1.2

Divide by .

Step 4.2

is approximately which is positive so remove the absolute value

Step 4.3

The exact value of is .

Step 5

Step 5.1

Move to the left of .

Step 5.2

The point is located in the first quadrant because and are both positive. The quadrants are labeled in counter-clockwise order, starting in the upper-right.

Quadrant

Quadrant

Step 6

is in the first quadrant.

Step 7

Use the formula to find the roots of the complex number.

,

Step 8

Step 8.1

Combine and .

Step 8.2

Combine and .

Step 8.3

Combine and .

Step 8.4

Combine and .

Step 8.5

Remove parentheses.

Step 8.5.1

Remove parentheses.

Step 8.5.2

Remove parentheses.

Step 8.5.3

Remove parentheses.

Step 8.5.4

Remove parentheses.

Step 8.5.5

Remove parentheses.

Step 8.5.6

Remove parentheses.

Step 8.5.7

Remove parentheses.

Step 8.5.8

Remove parentheses.

Step 9

Step 9.1

Rewrite as .

Step 9.2

Apply the power rule and multiply exponents, .

Step 9.3

Cancel the common factor of .

Step 9.3.1

Cancel the common factor.

Step 9.3.2

Rewrite the expression.

Step 9.4

Evaluate the exponent.

Step 9.5

Multiply .

Step 9.5.1

Multiply by .

Step 9.5.2

Multiply by .

Step 9.6

Add and .

Step 9.7

Multiply the numerator by the reciprocal of the denominator.

Step 9.8

Multiply .

Step 9.8.1

Multiply by .

Step 9.8.2

Multiply by .

Step 10

Step 10.1

Rewrite as .

Step 10.2

Apply the power rule and multiply exponents, .

Step 10.3

Cancel the common factor of .

Step 10.3.1

Cancel the common factor.

Step 10.3.2

Rewrite the expression.

Step 10.4

Evaluate the exponent.

Step 10.5

Multiply by .

Step 10.6

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

Step 10.7

Combine and .

Step 10.8

Combine the numerators over the common denominator.

Step 10.9

Simplify the numerator.

Step 10.9.1

Multiply by .

Step 10.9.2

Add and .

Step 10.10

Multiply the numerator by the reciprocal of the denominator.

Step 10.11

Multiply .

Step 10.11.1

Multiply by .

Step 10.11.2

Multiply by .

Step 11

Step 11.1

Rewrite as .

Step 11.2

Apply the power rule and multiply exponents, .

Step 11.3

Cancel the common factor of .

Step 11.3.1

Cancel the common factor.

Step 11.3.2

Rewrite the expression.

Step 11.4

Evaluate the exponent.

Step 11.5

Multiply by .

Step 11.6

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

Step 11.7

Combine and .

Step 11.8

Combine the numerators over the common denominator.

Step 11.9

Simplify the numerator.

Step 11.9.1

Multiply by .

Step 11.9.2

Add and .

Step 11.10

Multiply the numerator by the reciprocal of the denominator.

Step 11.11

Multiply .

Step 11.11.1

Multiply by .

Step 11.11.2

Multiply by .

Step 12

List the solutions.