Find Where Undefined/Discontinuous (1+cos(3t))/(sin(3t))+(sin(3t))/(1+cos(3t))

Find Where Undefined/Discontinuous (1+cos(3t))/(sin(3t))+(sin(3t))/(1+cos(3t))
Set the denominator in equal to to find where the expression is undefined.
Solve for .
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Take the inverse sine of both sides of the equation to extract from inside the sine.
The exact value of is .
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 .
Divide by .
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 the expression to find the second solution.
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Multiply by .
Add and .
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 .
Find the period.
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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 .
The period of the function is so values will repeat every radians in both directions.
, for any integer
Consolidate the answers.
, for any integer
, for any integer
Set the denominator in equal to to find where the expression is undefined.
Solve for .
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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 .
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 .
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 the expression to find the second solution.
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Subtract from .
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
Tap for more steps...
Cancel the common factor.
Divide by .
Find the period.
Tap for more steps...
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 .
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
The equation is undefined where the denominator equals , the argument of a square root is less than , or the argument of a logarithm is less than or equal to .
Set-Builder Notation:
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Name

Name one billion four hundred eleven million nine hundred forty-two thousand four hundred seventy-two

Interesting facts

  • 1411942472 has 32 divisors, whose sum is 4855218732
  • The reverse of 1411942472 is 2742491141
  • Previous prime number is 53

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 35
  • Digital Root 8

Name

Name one billion two hundred seventy-four million one hundred thousand nine hundred twenty-one

Interesting facts

  • 1274100921 has 16 divisors, whose sum is 1924569920
  • The reverse of 1274100921 is 1290014721
  • Previous prime number is 51

Basic properties

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

Name

Name one billion nine hundred sixty million eight hundred ten thousand seven hundred thirty-six

Interesting facts

  • 1960810736 has 128 divisors, whose sum is 10519773264
  • The reverse of 1960810736 is 6370180691
  • Previous prime number is 4861

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 41
  • Digital Root 5