Solve for x tan(x)+sec(x)=2cos(x)

Solve for x tan(x)+sec(x)=2cos(x)
Simplify each term.
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Rewrite in terms of sines and cosines.
Rewrite in terms of sines and cosines.
Find the LCD of the terms in the equation.
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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Since contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
The LCM is the smallest number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
The factor for is itself.
occurs time.
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Multiply each term by and simplify.
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Multiply each term in by in order to remove all the denominators from the equation.
Simplify each term.
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Multiply by by adding the exponents.
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Move .
Multiply by .
Since is on the right side of the equation, switch the sides so it is on the left side 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 root of both sides of the to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
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Simplify the right side of the equation.
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Combine the numerators over the common denominator.
Rewrite as .
Multiply by .
Combine and simplify the denominator.
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Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Combine using the product rule for radicals.
Reorder factors in .
The complete solution is the result of both the positive and negative portions of the solution.
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First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
Set up each of the solutions to solve for .
Set up the equation to solve for .
Solve the equation for .
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Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
Divide each term in the equation by .
Separate fractions.
Convert from to .
Divide by .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the left side.
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Rewrite in terms of sines and cosines.
Combine and .
Multiply the numerator by the reciprocal of the denominator.
Multiply and .
Move to the left of .
Cross multiply.
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Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.
Multiply by .
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Move all terms containing to the left side of the equation.
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Subtract from both sides of the equation.
Subtract from .
Multiply each term in by
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Multiply each term in by .
Multiply .
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Multiply by .
Multiply by .
Multiply by .
To remove the radical on the left side of the equation, square both sides of the equation.
Simplify each side of the equation.
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Simplify the left side of the equation.
Raising to any positive power yields .
Solve for .
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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 .
Subtract from 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 negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.
Simplify the expression to find the second solution.
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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 .
Add and .
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 .
Add and .
Subtract from .
The resulting angle of is positive, less than , and coterminal with .
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 .
Add to every negative angle to get positive angles.
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Add to to find the positive angle.
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 .
List the new angles.
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Set up the equation to solve for .
Solve the equation for .
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Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
Divide each term in the equation by .
Multiply the numerator by the reciprocal of the denominator.
Convert from to .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the left side.
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Rewrite in terms of sines and cosines.
Combine and .
Multiply the numerator by the reciprocal of the denominator.
Multiply and .
Move to the left of .
Cross multiply.
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Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.
Multiply by .
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Move all terms containing to the left side of the equation.
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Subtract from both sides of the equation.
Subtract from .
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 .
To remove the radical on the left side of the equation, square both sides of the equation.
Simplify each side of the equation.
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Simplify the left side of the equation.
Raising to any positive power yields .
Solve for .
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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 .
Subtract from 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 negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.
Simplify the expression to find the second solution.
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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 .
Add and .
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 .
Subtract from .
The resulting angle of is positive, less than , and coterminal with .
Find the period.
Tap for more steps...
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 .
Add to every negative angle to get positive angles.
Tap for more steps...
Add to to find the positive angle.
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...
Multiply by .
Subtract from .
List the new angles.
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
List all of the results found in the previous steps.
, for any integer
The complete solution is the set of all solutions.
, for any integer
Consolidate the answers.
, for any integer
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Name

Name eight hundred twenty-four million three hundred eighty thousand one hundred sixty-seven

Interesting facts

  • 824380167 has 32 divisors, whose sum is 1270600128
  • The reverse of 824380167 is 761083428
  • Previous prime number is 73

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 39
  • Digital Root 3

Name

Name one billion five hundred sixty-seven million six hundred thirty-one thousand seven hundred seventy-two

Interesting facts

  • 1567631772 has 64 divisors, whose sum is 4324561920
  • The reverse of 1567631772 is 2771367651
  • Previous prime number is 29

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 45
  • Digital Root 9

Name

Name one hundred eight million one hundred seventy thousand one hundred six

Interesting facts

  • 108170106 has 16 divisors, whose sum is 168461568
  • The reverse of 108170106 is 601071801
  • Previous prime number is 131

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 9
  • Sum of Digits 24
  • Digital Root 6