# Solve for x sin(2x)sin(x)-cos(2x)cos(x)=-cos(x)

Solve for x sin(2x)sin(x)-cos(2x)cos(x)=-cos(x)
Expand the trigonometric functions.
Simplify each term.
Apply the sine double-angle identity.
Multiply .
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 .
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Move to the left side of the equation by adding it to both sides.
Simplify the left side.
Simplify with factoring out.
Factor out of .
Factor out of .
Factor out of .
Multiply by .
Factor out of .
Factor out of .
Reorder and .
Apply pythagorean identity.
Simplify by adding terms.
Rewrite using the commutative property of multiplication.
Divide each term in by .
Cancel the common factor of .
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.
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 and solve.
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.
Simplify the right side of the equation.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
is equal to .
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 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.
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
The final solution is all the values that make true.
, for any integer
, for any integer
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### Name

Name one billion eight hundred six million eight hundred ninety-six thousand four hundred forty-two

### Interesting facts

• 1806896442 has 64 divisors, whose sum is 3107220480
• The reverse of 1806896442 is 2446986081
• Previous prime number is 31

### Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 48
• Digital Root 3

### Name

Name one hundred eighty-six million thirty thousand nine hundred forty-two

### Interesting facts

• 186030942 has 8 divisors, whose sum is 372061896
• The reverse of 186030942 is 249030681
• Previous prime number is 3

### Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 33
• Digital Root 6

### Name

Name one billion one hundred forty-six million one hundred sixty thousand three hundred nineteen

### Interesting facts

• 1146160319 has 4 divisors, whose sum is 1151079696
• The reverse of 1146160319 is 9130616411
• Previous prime number is 233

### Basic properties

• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 32
• Digital Root 5