# Trigonometry Examples

Find the Cube Roots of a Complex Number
,
Step 1
Calculate the distance from to the origin using the formula .
Step 2
Simplify .
Step 2.1
Use the power rule to distribute the exponent.
Step 2.1.1
Apply the product rule to .
Step 2.1.2
Apply the product rule to .
Step 2.1.3
Apply the product rule to .
Step 2.2
Simplify the expression.
Step 2.2.1
Raise to the power of .
Step 2.2.2
Multiply by .
Step 2.3
Simplify the numerator.
Step 2.3.1
Raise to the power of .
Step 2.3.2
Rewrite as .
Step 2.3.2.1
Use to rewrite as .
Step 2.3.2.2
Apply the power rule and multiply exponents, .
Step 2.3.2.3
Combine and .
Step 2.3.2.4
Cancel the common factor of .
Step 2.3.2.4.1
Cancel the common factor.
Step 2.3.2.4.2
Rewrite the expression.
Step 2.3.2.5
Evaluate the exponent.
Step 2.4
Reduce the expression by cancelling the common factors.
Step 2.4.1
Raise to the power of .
Step 2.4.2
Multiply by .
Step 2.4.3
Cancel the common factor of and .
Step 2.4.3.1
Factor out of .
Step 2.4.3.2
Cancel the common factors.
Step 2.4.3.2.1
Factor out of .
Step 2.4.3.2.2
Cancel the common factor.
Step 2.4.3.2.3
Rewrite the expression.
Step 2.5
Use the power rule to distribute the exponent.
Step 2.5.1
Apply the product rule to .
Step 2.5.2
Apply the product rule to .
Step 2.6
Simplify the numerator.
Step 2.6.1
Raise to the power of .
Step 2.6.2
Rewrite as .
Step 2.6.2.1
Use to rewrite as .
Step 2.6.2.2
Apply the power rule and multiply exponents, .
Step 2.6.2.3
Combine and .
Step 2.6.2.4
Cancel the common factor of .
Step 2.6.2.4.1
Cancel the common factor.
Step 2.6.2.4.2
Rewrite the expression.
Step 2.6.2.5
Evaluate the exponent.
Step 2.7
Reduce the expression by cancelling the common factors.
Step 2.7.1
Raise to the power of .
Step 2.7.2
Multiply by .
Step 2.7.3
Cancel the common factor of and .
Step 2.7.3.1
Factor out of .
Step 2.7.3.2
Cancel the common factors.
Step 2.7.3.2.1
Factor out of .
Step 2.7.3.2.2
Cancel the common factor.
Step 2.7.3.2.3
Rewrite the expression.
Step 2.7.4
Simplify the expression.
Step 2.7.4.1
Combine the numerators over the common denominator.
Step 2.7.4.2
Step 2.7.4.3
Divide by .
Step 2.7.4.4
Rewrite as .
Step 2.7.4.5
Pull terms out from under the radical, assuming positive real numbers.
Step 3
Calculate reference angle .
Step 4
Simplify .
Step 4.1
Cancel the common factor of .
Step 4.1.1
Cancel the common factor.
Step 4.1.2
Rewrite the expression.
Step 4.1.3
Move the negative one from the denominator of .
Step 4.2
Multiply by .
Step 4.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.4
The exact value of is .
Step 5
The point is located in the second quadrant because is negative and is positive. The quadrants are labeled in counter-clockwise order, starting in the upper-right.
Step 6
Step 7
Simplify .
Step 7.1
To write as a fraction with a common denominator, multiply by .
Step 7.2
Combine fractions.
Step 7.2.1
Combine and .
Step 7.2.2
Combine the numerators over the common denominator.
Step 7.3
Simplify the numerator.
Step 7.3.1
Move to the left of .
Step 7.3.2
Subtract from .
Step 8
Use the formula to find the roots of the complex number.
,
Step 9
Substitute , , and into the formula.
Step 9.1
To write as a fraction with a common denominator, multiply by .
Step 9.2
Combine and .
Step 9.3
Combine the numerators over the common denominator.
Step 9.4
Subtract from .
Step 9.4.1
Reorder and .
Step 9.4.2
Subtract from .
Step 9.5
Combine and .
Step 9.6
Combine and .
Step 9.7
Combine and .
Step 9.8
Combine and .
Step 9.9
Remove parentheses.
Step 9.9.1
Remove parentheses.
Step 9.9.2
Remove parentheses.
Step 9.9.3
Remove parentheses.
Step 9.9.4
Remove parentheses.
Step 9.9.5
Remove parentheses.
Step 9.9.6
Remove parentheses.
Step 9.9.7
Remove parentheses.
Step 10
Substitute into the formula and simplify.
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
To write as a fraction with a common denominator, multiply by .
Step 10.6
Combine and .
Step 10.7
Combine the numerators over the common denominator.
Step 10.8
Simplify the numerator.
Step 10.8.1
Move to the left of .
Step 10.8.2
Subtract from .
Step 10.9
Multiply .
Step 10.9.1
Multiply by .
Step 10.9.2
Multiply by .
Step 10.10
Step 10.11
Multiply the numerator by the reciprocal of the denominator.
Step 10.12
Cancel the common factor of .
Step 10.12.1
Factor out of .
Step 10.12.2
Cancel the common factor.
Step 10.12.3
Rewrite the expression.
Step 11
Substitute into the formula and simplify.
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
To write as a fraction with a common denominator, multiply by .
Step 11.6
Combine and .
Step 11.7
Combine the numerators over the common denominator.
Step 11.8
Simplify the numerator.
Step 11.8.1
Move to the left of .
Step 11.8.2
Subtract from .
Step 11.9
Multiply by .
Step 11.10
To write as a fraction with a common denominator, multiply by .
Step 11.11
Combine and .
Step 11.12
Combine the numerators over the common denominator.
Step 11.13
Simplify the numerator.
Step 11.13.1
Multiply by .
Step 11.13.2
Step 11.14
Multiply the numerator by the reciprocal of the denominator.
Step 11.15
Multiply .
Step 11.15.1
Multiply by .
Step 11.15.2
Multiply by .
Step 12
Substitute into the formula and simplify.
Step 12.1
Rewrite as .
Step 12.2
Apply the power rule and multiply exponents, .
Step 12.3
Cancel the common factor of .
Step 12.3.1
Cancel the common factor.
Step 12.3.2
Rewrite the expression.
Step 12.4
Evaluate the exponent.
Step 12.5
To write as a fraction with a common denominator, multiply by .
Step 12.6
Combine and .
Step 12.7
Combine the numerators over the common denominator.
Step 12.8
Simplify the numerator.
Step 12.8.1
Move to the left of .
Step 12.8.2
Subtract from .
Step 12.9
Multiply by .
Step 12.10
To write as a fraction with a common denominator, multiply by .
Step 12.11
Combine and .
Step 12.12
Combine the numerators over the common denominator.
Step 12.13
Simplify the numerator.
Step 12.13.1
Multiply by .
Step 12.13.2
Step 12.14
Multiply the numerator by the reciprocal of the denominator.
Step 12.15
Multiply .
Step 12.15.1
Multiply by .
Step 12.15.2
Multiply by .
Step 13
List the solutions.