Solve for x in Radians sin(2x)-sin(x)=0

Solve for x in Radians sin(2x)-sin(x)=0
Apply the sine double-angle identity.
Factor out of .
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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 equal to and solve for .
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Set equal to .
Solve for .
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Take the inverse sine of both sides of the equation to extract from inside the sine.
Simplify the right side.
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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 of .
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The period of the function can be calculated using .
Replace with in the formula for period.
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
, for any integer
Set equal to and solve for .
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Set equal to .
Solve for .
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Add to both sides of the equation.
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Simplify the right side.
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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 .
Combine fractions.
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Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Find the period of .
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The period of the function can be calculated using .
Replace with in the formula for period.
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
, 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 four hundred thirteen million eighty-six thousand nine hundred ninety-three

Interesting facts

  • 413086993 has 4 divisors, whose sum is 450640368
  • The reverse of 413086993 is 399680314
  • Previous prime number is 11

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 43
  • Digital Root 7

Name

Name one billion three hundred fifty-nine million six hundred eighty-seven thousand seven hundred sixty-five

Interesting facts

  • 1359687765 has 8 divisors, whose sum is 1500345600
  • The reverse of 1359687765 is 5677869531
  • Previous prime number is 29

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 10
  • Sum of Digits 57
  • Digital Root 3

Name

Name one billion nine hundred eighty-six million four hundred twelve thousand six hundred sixty-one

Interesting facts

  • 1986412661 has 8 divisors, whose sum is 1999005624
  • The reverse of 1986412661 is 1662146891
  • Previous prime number is 853

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 10
  • Sum of Digits 44
  • Digital Root 8