# Solve for x sin(2x)+cos(x)=0

Factor out of .

Raise to the power of .

Factor out of .

Factor out of .

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 .

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 .

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 .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Subtract from .

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 .

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 .

Subtract from both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Move the negative in front of the fraction.

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

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.

Simplify .

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 .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Add and .

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 .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of .

Add and .

Subtract from .

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

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 .

Add to every negative angle to get positive angles.

Add to to find the positive angle.

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

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

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

The final solution is all the values that make true.

, for any integer

Consolidate and to .

, for any integer

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