# Trigonometry Examples

Find All Complex Number Solutions
Substitute for .
This is the trigonometric form of a complex number where is the modulus and is the angle created on the complex plane.
The modulus of a complex number is the distance from the origin on the complex plane.
where
Substitute the actual values of and .
Pull terms out from under the radical, assuming positive real numbers.
The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.
Since the argument is undefined and is positive, the angle of the point on the complex plane is .
Substitute the values of and .
Replace the right side of the equation with the trigonometric form.
Use De Moivre's Theorem to find an equation for .
Equate the modulus of the trigonometric form to to find the value of .
Solve the equation for .
Take the root of both sides of the to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
Find the approximate value of .
Find the possible values of .
and
Finding all the possible values of leads to the equation .
Find the value of for .
Solve the equation for .
Multiply .
Multiply by .
For
Multiply by .
For
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Multiply .
Multiply and .
Multiply by .
Use the values of and to find a solution to the equation .
Convert the solution to rectangular form.
Simplify each term.
The exact value of is .
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the cosine half-angle identity .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Change the to because cosine is positive in the first quadrant.
Factor out of .
Cancel the common factor.
Rewrite the expression.
The exact value of is .
Simplify .
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Multiply the numerator by the reciprocal of the denominator.
Multiply .
Multiply and .
Multiply by .
Rewrite as .
Simplify the denominator.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
The exact value of is .
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the sine half-angle identity.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Change the to because sine is positive in the first quadrant.
Factor out of .
Cancel the common factor.
Rewrite the expression.
The exact value of is .
Simplify .
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Multiply the numerator by the reciprocal of the denominator.
Multiply .
Multiply and .
Multiply by .
Rewrite as .
Simplify the denominator.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Combine and .
Simplify terms.
Combine the numerators over the common denominator.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Multiply by .
Multiply by .
Substitute for to calculate the value of after the right shift.
Find the value of for .
Solve the equation for .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Multiply by .
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Multiply .
Multiply and .
Multiply by .
Use the values of and to find a solution to the equation .
Convert the solution to rectangular form.
Simplify each term.
The exact value of is .
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the cosine half-angle identity .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Change the to because cosine is negative in the second quadrant.
Simplify .
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.
The exact value of is .
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Multiply the numerator by the reciprocal of the denominator.
Multiply .
Multiply and .
Multiply by .
Rewrite as .
Simplify the denominator.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
The exact value of is .
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the sine half-angle identity.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Change the to because sine is positive in the second quadrant.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify .
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.
The exact value of is .
Multiply .
Multiply by .
Multiply by .
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Multiply the numerator by the reciprocal of the denominator.
Multiply .
Multiply and .
Multiply by .
Rewrite as .
Simplify the denominator.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Combine and .
Simplify terms.
Combine the numerators over the common denominator.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Multiply .
Multiply by .
Multiply by .
Multiply by .
Substitute for to calculate the value of after the right shift.
Find the value of for .
Solve the equation for .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Multiply by .
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Multiply .
Multiply and .
Multiply by .
Use the values of and to find a solution to the equation .
Convert the solution to rectangular form.
Simplify each term.
The exact value of is .
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the cosine half-angle identity .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Change the to because cosine is negative in the third quadrant.
Simplify .
Subtract full rotations of until the angle is greater than or equal to and less than .
The exact value of is .
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Multiply the numerator by the reciprocal of the denominator.
Multiply .
Multiply and .
Multiply by .
Rewrite as .
Simplify the denominator.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
The exact value of is .
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the sine half-angle identity.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Change the to because sine is negative in the third quadrant.
Simplify .
Subtract full rotations of until the angle is greater than or equal to and less than .
The exact value of is .
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Multiply the numerator by the reciprocal of the denominator.
Multiply .
Multiply and .
Multiply by .
Rewrite as .
Simplify the denominator.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Combine and .
Simplify terms.
Combine the numerators over the common denominator.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Multiply .
Multiply by .
Multiply by .
Multiply .
Multiply by .
Multiply by .
Substitute for to calculate the value of after the right shift.
Find the value of for .
Solve the equation for .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Multiply by .
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Multiply .
Multiply and .
Multiply by .
Use the values of and to find a solution to the equation .
Convert the solution to rectangular form.
Simplify each term.
The exact value of is .
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the cosine half-angle identity .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Change the to because cosine is positive in the fourth quadrant.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify .
Subtract full rotations of until the angle is greater than or equal to and less than .
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.
The exact value of is .
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Multiply the numerator by the reciprocal of the denominator.
Multiply .
Multiply and .
Multiply by .
Rewrite as .
Simplify the denominator.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
The exact value of is .
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the sine half-angle identity.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Change the to because sine is negative in the fourth quadrant.
Simplify .
Subtract full rotations of until the angle is greater than or equal to and less than .
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.
The exact value of is .
Multiply .
Multiply by .
Multiply by .
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Multiply the numerator by the reciprocal of the denominator.
Multiply .
Multiply and .
Multiply by .
Rewrite as .
Simplify the denominator.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Combine and .
Simplify terms.
Combine the numerators over the common denominator.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Multiply by .
Multiply .
Multiply by .
Multiply by .
Substitute for to calculate the value of after the right shift.
These are the complex solutions to .