Solve for x sin(4x)=-2sin(2x)

Math
Rewrite the equation as .
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Move the negative in front of the fraction.
Move all terms containing to the left side of the equation.
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Add to both sides of the equation.
Simplify the left side of the equation.
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Simplify each term.
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Apply the sine doubleangle identity.
Simplify the numerator.
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Factor out of .
Apply the sine doubleangle identity.
Use the doubleangle identity to transform to .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Multiply by .
Simplify the numerator.
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Rewrite.
Factor out of .
Apply the sine doubleangle identity.
Use the doubleangle identity to transform to .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Multiply by .
Remove unnecessary parentheses.
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Apply the sine doubleangle identity.
Apply the distributive property.
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Combine the opposite terms in .
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Reorder the factors in the terms and .
Add and .
Add and .
Simplify each term.
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Multiply .
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Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite using the commutative property of multiplication.
Multiply .
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Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Apply the sine doubleangle identity.
Move .
Multiply by .
Simplify with factoring out.
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Factor out of .
Factor out of .
Apply pythagorean identity.
Add and .
Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Divide by .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to and solve.
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Set the first factor equal to .
Take the inverse sine of both sides of the equation to extract from inside the sine.
The exact value of is .
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Divide by .
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 the expression to find the second solution.
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Multiply by .
Add and .
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Find the period.
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The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
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The absolute value is the distance between a number and zero. The distance between and is .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Set the next factor equal to and solve.
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Set the next factor equal to .
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.
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Simplify the right side of the equation.
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Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
is equal to .
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
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 .
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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 .
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Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Find the period.
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The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
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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
The final solution is all the values that make true.
, for any integer
Consolidate the answers.
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Consolidate and to .
, for any integer
Consolidate and to .
, for any integer
Consolidate and to .
, for any integer
, for any integer
Solve for x sin(4x)=-2sin(2x)

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