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
Solve the equation for .
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Take the 4th root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
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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.
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 .
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Simplify.
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Multiply .
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Multiply by .
Multiply by .
Add and .
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Multiply the numerator by the reciprocal of the denominator.
Multiply .
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Multiply by .
Multiply by .
Step 18
Use the values of and to find a solution to the equation .
Step 19
Convert the solution to rectangular form.
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Simplify each term.
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The exact value of is .
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Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the cosine half-angle identity .
Change the to because cosine is positive in the first quadrant.
The exact value of is .
Simplify .
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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 .
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Multiply by .
Multiply by .
Rewrite as .
Simplify the denominator.
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Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
The exact value of is .
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Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the sine half-angle identity.
Change the to because sine is positive in the first quadrant.
Simplify .
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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 .
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Multiply by .
Multiply by .
Rewrite as .
Simplify the denominator.
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Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Combine and .
Simplify terms.
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Combine the numerators over the common denominator.
Combine and .
Factor out of .
Separate fractions.
Simplify the expression.
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Divide by .
Divide by .
Apply the distributive property.
Multiply by .
Multiply 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 .
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Simplify.
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Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Multiply by .
Add and .
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Multiply the numerator by the reciprocal of the denominator.
Multiply .
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Multiply by .
Multiply by .
Step 23
Use the values of and to find a solution to the equation .
Step 24
Convert the solution to rectangular form.
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Simplify each term.
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The exact value of is .
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Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the cosine half-angle identity .
Change the to because cosine is negative in the second quadrant.
Simplify .
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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 .
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Multiply by .
Multiply by .
Rewrite as .
Simplify the denominator.
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Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
The exact value of is .
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Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the sine half-angle identity.
Change the to because sine is positive in the second quadrant.
Simplify .
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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 .
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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 .
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Multiply by .
Multiply by .
Rewrite as .
Simplify the denominator.
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Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Combine and .
Simplify terms.
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Combine the numerators over the common denominator.
Combine and .
Factor out of .
Separate fractions.
Simplify the expression.
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Divide by .
Divide by .
Apply the distributive property.
Multiply .
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Multiply by .
Multiply by .
Multiply 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 .
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Simplify.
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Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Multiply by .
Add and .
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Multiply the numerator by the reciprocal of the denominator.
Multiply .
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Multiply by .
Multiply by .
Step 28
Use the values of and to find a solution to the equation .
Step 29
Convert the solution to rectangular form.
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Simplify each term.
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The exact value of is .
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Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the cosine half-angle identity .
Change the to because cosine is negative in the third quadrant.
Simplify .
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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 .
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Multiply by .
Multiply by .
Rewrite as .
Simplify the denominator.
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Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
The exact value of is .
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Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the sine half-angle identity.
Change the to because sine is negative in the third quadrant.
Simplify .
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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 .
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Multiply by .
Multiply by .
Rewrite as .
Simplify the denominator.
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Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Combine and .
Simplify terms.
Tap for more steps...
Combine the numerators over the common denominator.
Combine and .
Factor out of .
Separate fractions.
Simplify the expression.
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Divide by .
Divide by .
Apply the distributive property.
Multiply .
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Multiply by .
Multiply by .
Multiply .
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Multiply by .
Multiply by .
Step 30
Substitute for to calculate the value of after the right shift.
Step 31
Find the value of for .
Step 32
Solve the equation for .
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Simplify.
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Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Multiply by .
Add and .
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Multiply the numerator by the reciprocal of the denominator.
Multiply .
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Multiply by .
Multiply by .
Step 33
Use the values of and to find a solution to the equation .
Step 34
Convert the solution to rectangular form.
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Simplify each term.
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The exact value of is .
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Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the cosine half-angle identity .
Change the to because cosine is positive in the fourth quadrant.
Simplify .
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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 .
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Multiply by .
Multiply by .
Rewrite as .
Simplify the denominator.
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Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
The exact value of is .
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Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Apply the sine half-angle identity.
Change the to because sine is negative in the fourth quadrant.
Simplify .
Tap for more steps...
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 .
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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 .
Tap for more steps...
Multiply by .
Multiply by .
Rewrite as .
Simplify the denominator.
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Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Combine and .
Simplify terms.
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Combine the numerators over the common denominator.
Combine and .
Factor out of .
Separate fractions.
Simplify the expression.
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Divide by .
Divide by .
Apply the distributive property.
Multiply by .
Multiply .
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Multiply by .
Multiply by .
Step 35
Substitute for to calculate the value of after the right shift.
Step 36
These are the complex solutions to .
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