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

Solve for x in Radians cos(2x)-cos(x)=0
Use the double-angle identity to transform to .
Factor by grouping.
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Reorder terms.
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Factor out of .
Rewrite as plus
Apply the distributive property.
Multiply by .
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .
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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Subtract from 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 .
Simplify the right side.
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Move the negative in front of the fraction.
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 negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third 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
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.
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.
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 the answers.
, for any integer
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Name

Name one hundred twenty-seven million six hundred nineteen thousand two hundred thirty-two

Interesting facts

  • 127619232 has 256 divisors, whose sum is 1391554080
  • The reverse of 127619232 is 232916721
  • Previous prime number is 13

Basic properties

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

Name

Name one hundred seven million one hundred ninety-five thousand three hundred ninety-two

Interesting facts

  • 107195392 has 2048 divisors, whose sum is 6181485516
  • The reverse of 107195392 is 293591701
  • Previous prime number is 2

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 9
  • Sum of Digits 37
  • Digital Root 1

Name

Name four hundred four million seven hundred seventy-three thousand five hundred ninety-five

Interesting facts

  • 404773595 has 16 divisors, whose sum is 489900960
  • The reverse of 404773595 is 595377404
  • Previous prime number is 733

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 44
  • Digital Root 8