# Trigonometry Examples

Find the Fourth Roots of a Complex Number
,
Step 1
Calculate the distance from to the origin using the formula .
Step 2
Simplify .
Step 2.1
Simplify the expression.
Step 2.1.1
Apply the product rule to .
Step 2.1.2
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
Raise to the power of .
Step 2.3.3
Step 2.3.4
Rewrite as .
Step 2.3.5
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 4.2
Multiply by .
Step 4.3
Combine and simplify the denominator.
Step 4.3.1
Multiply by .
Step 4.3.2
Raise to the power of .
Step 4.3.3
Raise to the power of .
Step 4.3.4
Use the power rule to combine exponents.
Step 4.3.5
Step 4.3.6
Rewrite as .
Step 4.3.6.1
Use to rewrite as .
Step 4.3.6.2
Apply the power rule and multiply exponents, .
Step 4.3.6.3
Combine and .
Step 4.3.6.4
Cancel the common factor of .
Step 4.3.6.4.1
Cancel the common factor.
Step 4.3.6.4.2
Rewrite the expression.
Step 4.3.6.5
Evaluate the exponent.
Step 4.4
is approximately which is positive so remove the absolute value
Step 4.5
The exact value of is .
Step 5
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.
Step 6
Step 7
Use the formula to find the roots of the complex number.
,
Step 8
Substitute , , and into the formula.
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
Substitute into the formula and simplify.
Step 9.1
Remove parentheses.
Step 9.2
Multiply .
Step 9.2.1
Multiply by .
Step 9.2.2
Multiply by .
Step 9.3
Step 9.4
Multiply the numerator by the reciprocal of the denominator.
Step 9.5
Multiply .
Step 9.5.1
Multiply by .
Step 9.5.2
Multiply by .
Step 10
Substitute into the formula and simplify.
Step 10.1
Remove parentheses.
Step 10.2
Multiply by .
Step 10.3
To write as a fraction with a common denominator, multiply by .
Step 10.4
Combine and .
Step 10.5
Combine the numerators over the common denominator.
Step 10.6
Simplify the numerator.
Step 10.6.1
Multiply by .
Step 10.6.2
Step 10.7
Multiply the numerator by the reciprocal of the denominator.
Step 10.8
Multiply .
Step 10.8.1
Multiply by .
Step 10.8.2
Multiply by .
Step 11
Substitute into the formula and simplify.
Step 11.1
Remove parentheses.
Step 11.2
Multiply by .
Step 11.3
To write as a fraction with a common denominator, multiply by .
Step 11.4
Combine and .
Step 11.5
Combine the numerators over the common denominator.
Step 11.6
Simplify the numerator.
Step 11.6.1
Multiply by .
Step 11.6.2
Step 11.7
Multiply the numerator by the reciprocal of the denominator.
Step 11.8
Multiply .
Step 11.8.1
Multiply by .
Step 11.8.2
Multiply by .
Step 12
Substitute into the formula and simplify.
Step 12.1
Remove parentheses.
Step 12.2
Multiply by .
Step 12.3
To write as a fraction with a common denominator, multiply by .
Step 12.4
Combine and .
Step 12.5
Combine the numerators over the common denominator.
Step 12.6
Simplify the numerator.
Step 12.6.1
Multiply by .
Step 12.6.2