Solve for x 10cos(x)^2+5cos(x)-5=0

Solve for x 10cos(x)^2+5cos(x)-5=0
Factor the left side of the equation.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Let . Substitute for all occurrences of .
Factor by grouping.
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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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Multiply by .
Rewrite as plus
Apply the distributive property.
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, .
Factor.
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Replace all occurrences of with .
Remove unnecessary parentheses.
Replace the left side with the factored expression.
Divide each term in by .
Cancel the common factor of .
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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.
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Set the first factor equal to .
Add to both sides of the equation.
Divide each term by and simplify.
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Divide each term in by .
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.
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 .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Find the period.
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The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
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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.
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Set the next factor equal to .
Subtract from both sides of the equation.
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 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.
Subtract from .
Find the period.
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The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
Tap for more steps...
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 the answers.
, for any integer
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Name

Name one billion seven hundred forty-six million sixty-four thousand one hundred fifty-nine

Interesting facts

  • 1746064159 has 16 divisors, whose sum is 2084705280
  • The reverse of 1746064159 is 9514606471
  • Previous prime number is 919

Basic properties

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

Name

Name seven million one hundred eighty-four thousand three hundred twenty-five

Interesting facts

  • 7184325 has 4 divisors, whose sum is 7280192
  • The reverse of 7184325 is 5234817
  • Previous prime number is 75

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 7
  • Sum of Digits 30
  • Digital Root 3

Name

Name four hundred eighty-six million three hundred fifty-two thousand eight hundred ninety-seven

Interesting facts

  • 486352897 has 4 divisors, whose sum is 487011760
  • The reverse of 486352897 is 798253684
  • Previous prime number is 739

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 52
  • Digital Root 7