# Solve for x 2sin(2x)cos(x)+2cos(2x)sin(x) = square root of 3

Solve for x 2sin(2x)cos(x)+2cos(2x)sin(x) = square root of 3
Square both sides of the equation.
Simplify terms.
Simplify each term.
Apply the sine double-angle identity.
Multiply by to get .
Remove parentheses around .
Simplify .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Use the double-angle identity to transform to .
Apply the distributive property.
Multiply by to get .
Apply the distributive property.
Move .
Use the power rule to combine exponents.
Remove unnecessary parentheses.
Reorder terms.
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Remove parentheses.
Simplify and combine like terms.
Simplify each term.
Move .
Use the power rule to combine exponents.
Simplify .
Multiply by to get .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Move .
Use the power rule to combine exponents.
Multiply by to get .
Move .
Use the power rule to combine exponents.
Multiply by to get .
Remove parentheses around .
Move .
Use the power rule to combine exponents.
Multiply by to get .
Subtract from to get .
Remove parentheses around .
Rewrite as .
Replace the with based on the identity.
Simplify each term.
Move .
Use the power rule to combine exponents.
Reorder the polynomial alphabetically from left to right, starting with the highest order term.
Move all terms not containing to the right side of the equation.
Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.
Substitute into the equation. This will make the quadratic formula easy to use.
Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.
Move all the expressions to the left side of the equation.
Move to the left side of the equation by adding it to both sides.
Move to the left side of the equation by subtracting it from both sides.
Substitute the real value of back into the solved equation.
Take the square 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.
Simplify the right side of the equation.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
is equal to .
Simplify the expression to find the first solution.
Take the inverse of both sides of the equation to extract from inside the .
The exact value of is .
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Simplify the expression to find the second solution.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by to get .
Combine the numerators over the common denominator.
Simplify the numerator.
Factor out of .
Multiply by to get .
Subtract from to get .
Simplify the expression.
Move to the left of the expression .
Multiply by to get .
Find the period.
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
The absolute value is the distance between a number and zero. The distance between and is .
Divide by to get .
The period of the function is so values will repeat every radians in both directions.