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

Solve for x sin(2x)=sin(x)
Factor out of .
Factor out 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 and solve.
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 .
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
Set the next factor equal to and solve.
Set the next factor equal to .
Add to 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 .
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
The final solution is all the values that make true.
, for any integer
Consolidate and to .
, for any integer
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### Name

Name one billion one hundred forty-five million eleven thousand six hundred thirty

### Interesting facts

• 1145011630 has 16 divisors, whose sum is 1859569920
• The reverse of 1145011630 is 0361105411
• Previous prime number is 19

### Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 22
• Digital Root 4

### Name

Name one billion three hundred twenty-nine million three hundred eighty-eight thousand five hundred thirty-five

### Interesting facts

• 1329388535 has 16 divisors, whose sum is 1606318560
• The reverse of 1329388535 is 5358839231
• Previous prime number is 419

### Basic properties

• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 47
• Digital Root 2

### Name

Name one billion four hundred fifty-four million five hundred sixty-five thousand three hundred thirty-seven

### Interesting facts

• 1454565337 has 4 divisors, whose sum is 1454842420
• The reverse of 1454565337 is 7335654541
• Previous prime number is 5353

### Basic properties

• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 43
• Digital Root 7