# Find All Complex Solutions 2cos(x)^2+sin(x)-1=0

Replace the with based on the identity.

Simplify each term.

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Apply the distributive property.

Multiply by .

Multiply by .

Subtract from .

Substitute for .

Factor the left side of the equation.

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Factor out of .

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Factor out of .

Factor out of .

Rewrite as .

Factor out of .

Factor out of .

Factor.

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Factor by grouping.

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For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .

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Factor out of .

Rewrite as plus

Apply the distributive property.

Multiply by .

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

Factor out the greatest common factor (GCF) from each group.

Factor the polynomial by factoring out the greatest common factor, .

Remove unnecessary parentheses.

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Set equal to and solve for .

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Set equal to .

Solve for .

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Subtract from both sides of the equation.

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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Move the negative in front of the fraction.

Set equal to and solve for .

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Set equal to .

Add to both sides of the equation.

The final solution is all the values that make true.

Substitute for .

Set up each of the solutions to solve for .

Solve for in .

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Take the inverse sine of both sides of the equation to extract from inside the sine.

Simplify the right side.

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The exact value of is .

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.

Simplify the expression to find the second solution.

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Subtract from .

The resulting angle of is positive, less than , and coterminal with .

Find the period of .

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The period of the function can be calculated using .

Replace with in the formula for period.

The absolute value is the distance between a number and zero. The distance between and is .

Divide by .

Add to every negative angle to get positive angles.

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Add to to find the positive angle.

To write as a fraction with a common denominator, multiply by .

Combine fractions.

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Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

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Multiply by .

Subtract from .

List the new angles.

The period of the function is so values will repeat every radians in both directions.

, for any integer

, for any integer

Solve for in .

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Take the inverse sine of both sides of the equation to extract from inside the sine.

Simplify the right side.

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The exact value of is .

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.

Simplify .

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To write as a fraction with a common denominator, multiply by .

Combine fractions.

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Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

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Move to the left of .

Subtract from .

Find the period of .

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The period of the function can be calculated using .

Replace with in the formula for period.

The absolute value is the distance between a number and zero. The distance between and is .

Divide by .

The period of the function is so values will repeat every radians in both directions.

, for any integer

, for any integer

List all of the solutions.

, for any integer

Consolidate the answers.

, for any integer

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