Solve over the Interval sin(2x)-2cos(x)=0 , (0,2pi)

Solve over the Interval sin(2x)-2cos(x)=0 , (0,2pi)
,
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
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 .
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 .
Add to both sides of the equation.
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.
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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Move to the left of .
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
Find the values of that produce a value within the interval .
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Plug in for and simplify to see if the solution is contained in .
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Plug in for .
Simplify.
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Multiply by .
Add and .
The interval contains .
Plug in for and simplify to see if the solution is contained in .
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Plug in for .
Simplify.
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Multiply by .
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 .
Tap for more steps...
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps...
Move to the left of .
Add and .
The interval contains .
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Name

Name one billion forty-five million eighty-one thousand seven hundred ten

Interesting facts

  • 1045081710 has 16 divisors, whose sum is 2090756304
  • The reverse of 1045081710 is 0171805401
  • Previous prime number is 3

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 27
  • Digital Root 9

Name

Name eight hundred thirty-eight million two hundred twenty-three thousand seven hundred ninety-four

Interesting facts

  • 838223794 has 8 divisors, whose sum is 1331296668
  • The reverse of 838223794 is 497322838
  • Previous prime number is 17

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 9
  • Sum of Digits 46
  • Digital Root 1

Name

Name one billion eleven million five hundred thirty-nine thousand sixty

Interesting facts

  • 1011539060 has 32 divisors, whose sum is 3121320960
  • The reverse of 1011539060 is 0609351101
  • Previous prime number is 7

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 26
  • Digital Root 8