# Trigonometry Examples

Find All Complex Number Solutions
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
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
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
Solve the equation for .
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
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
Convert the solution to rectangular form.
Step 19.1
Simplify each term.
Step 19.1.1
The exact value of is .
Step 19.1.2
The exact value of is .
Step 19.1.3
Combine and .
Step 19.2
Apply the distributive property.
Step 19.3
Multiply .
Step 19.3.1
Combine and .
Step 19.3.2
Multiply by .
Step 19.4
Combine and .
Step 19.5
Simplify each term.
Step 19.5.1
Divide by .
Step 19.5.2
Factor out of .
Step 19.5.3
Factor out of .
Step 19.5.4
Separate fractions.
Step 19.5.5
Divide by .
Step 19.5.6
Divide by .
Step 20
Substitute for to calculate the value of after the right shift.
Step 21
Find the value of for .
Step 22
Solve the equation for .
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
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
Convert the solution to rectangular form.
Step 24.1
Simplify each term.
Step 24.1.1
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 second quadrant.
Step 24.1.2
The exact value of is .
Step 24.1.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 24.1.4
The exact value of is .
Step 24.1.5
Combine and .
Step 24.2
Apply the distributive property.
Step 24.3
Multiply .
Step 24.3.1
Multiply by .
Step 24.3.2
Combine and .
Step 24.3.3
Multiply by .
Step 24.4
Combine and .
Step 24.5
Simplify each term.
Step 24.5.1
Divide by .
Step 24.5.2
Factor out of .
Step 24.5.3
Factor out of .
Step 24.5.4
Separate fractions.
Step 24.5.5
Divide by .
Step 24.5.6
Divide by .
Step 25
Substitute for to calculate the value of after the right shift.
Step 26
Find the value of for .
Step 27
Solve the equation for .
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
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
Cancel the common factor of .
Step 27.2.3.2.1
Factor out of .
Step 27.2.3.2.2
Cancel the common factor.
Step 27.2.3.2.3
Rewrite the expression.
Step 28
Use the values of and to find a solution to the equation .
Step 29
Convert the solution to rectangular form.
Step 29.1
Simplify each term.
Step 29.1.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 29.1.2
The exact value of is .
Step 29.1.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.
Step 29.1.4
The exact value of is .
Step 29.1.5
Multiply by .
Step 29.1.6
Move to the left of .
Step 29.1.7
Rewrite as .
Step 29.2
Simplify the expression.
Step 29.2.1
Subtract from .
Step 29.2.2
Multiply by .
Step 30
Substitute for to calculate the value of after the right shift.
Step 31
These are the complex solutions to .