Topics

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

Replace the with based on the identity.
Simplify each term.
Click for detailed explanation...
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Substitute for .
Factor the left side of the equation.
Click for detailed explanation...
Factor out of .
Click for detailed explanation...
Factor out of .
Factor out of .
Rewrite as .
Factor out of .
Factor out of .
Factor.
Click for detailed explanation...
Factor by grouping.
Click for detailed explanation...
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Click for detailed explanation...
Factor out of .
Rewrite as plus
Apply the distributive property.
Multiply by .
Factor out the greatest common factor from each group.
Click for detailed explanation...
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 .
Click for detailed explanation...
Set equal to .
Solve for .
Click for detailed explanation...
Subtract from both sides of the equation.
Divide each term in by and simplify.
Click for detailed explanation...
Divide each term in by .
Simplify the left side.
Click for detailed explanation...
Cancel the common factor of .
Click for detailed explanation...
Cancel the common factor.
Divide by .
Simplify the right side.
Click for detailed explanation...
Move the negative in front of the fraction.
Set equal to and solve for .
Click for detailed explanation...
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 .
Click for detailed explanation...
Take the inverse sine of both sides of the equation to extract from inside the sine.
Simplify the right side.
Click for detailed explanation...
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.
Click for detailed explanation...
Subtract from .
The resulting angle of is positive, less than , and coterminal with .
Find the period of .
Click for detailed explanation...
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.
Click for detailed explanation...
Add to to find the positive angle.
To write as a fraction with a common denominator, multiply by .
Combine fractions.
Click for detailed explanation...
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Click for detailed explanation...
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 .
Click for detailed explanation...
Take the inverse sine of both sides of the equation to extract from inside the sine.
Simplify the right side.
Click for detailed explanation...
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 .
Click for detailed explanation...
To write as a fraction with a common denominator, multiply by .
Combine fractions.
Click for detailed explanation...
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Click for detailed explanation...
Move to the left of .
Subtract from .
Find the period of .
Click for detailed explanation...
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