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Finite Math Examples
Step 1
Set the denominator in equal to to find where the expression is undefined.
Step 2
Step 2.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.2
Set equal to and solve for .
Step 2.2.1
Set equal to .
Step 2.2.2
Solve for .
Step 2.2.2.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2.2.2.2
Simplify the right side.
Step 2.2.2.2.1
The exact value of is .
Step 2.2.2.3
Divide each term in by and simplify.
Step 2.2.2.3.1
Divide each term in by .
Step 2.2.2.3.2
Simplify the left side.
Step 2.2.2.3.2.1
Cancel the common factor of .
Step 2.2.2.3.2.1.1
Cancel the common factor.
Step 2.2.2.3.2.1.2
Divide by .
Step 2.2.2.3.3
Simplify the right side.
Step 2.2.2.3.3.1
Divide by .
Step 2.2.2.4
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.
Step 2.2.2.5
Solve for .
Step 2.2.2.5.1
Simplify.
Step 2.2.2.5.1.1
Multiply by .
Step 2.2.2.5.1.2
Add and .
Step 2.2.2.5.2
Divide each term in by and simplify.
Step 2.2.2.5.2.1
Divide each term in by .
Step 2.2.2.5.2.2
Simplify the left side.
Step 2.2.2.5.2.2.1
Cancel the common factor of .
Step 2.2.2.5.2.2.1.1
Cancel the common factor.
Step 2.2.2.5.2.2.1.2
Divide by .
Step 2.2.2.5.2.3
Simplify the right side.
Step 2.2.2.5.2.3.1
Divide by .
Step 2.2.2.6
Find the period of .
Step 2.2.2.6.1
The period of the function can be calculated using .
Step 2.2.2.6.2
Replace with in the formula for period.
Step 2.2.2.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.2.2.6.4
Divide by .
Step 2.2.2.7
The period of the function is so values will repeat every degrees in both directions.
, for any integer
, for any integer
, for any integer
Step 2.3
Set equal to and solve for .
Step 2.3.1
Set equal to .
Step 2.3.2
Solve for .
Step 2.3.2.1
Simplify the left side.
Step 2.3.2.1.1
Simplify each term.
Step 2.3.2.1.1.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 2.3.2.1.1.2
The exact value of is .
Step 2.3.2.2
Subtract from both sides of the equation.
Step 2.3.2.3
Divide each term in by and simplify.
Step 2.3.2.3.1
Divide each term in by .
Step 2.3.2.3.2
Simplify the left side.
Step 2.3.2.3.2.1
Cancel the common factor of .
Step 2.3.2.3.2.1.1
Cancel the common factor.
Step 2.3.2.3.2.1.2
Divide by .
Step 2.3.2.3.3
Simplify the right side.
Step 2.3.2.3.3.1
Dividing two negative values results in a positive value.
Step 2.3.2.4
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.3.2.5
Simplify .
Step 2.3.2.5.1
Rewrite as .
Step 2.3.2.5.2
Any root of is .
Step 2.3.2.5.3
Multiply by .
Step 2.3.2.5.4
Combine and simplify the denominator.
Step 2.3.2.5.4.1
Multiply by .
Step 2.3.2.5.4.2
Raise to the power of .
Step 2.3.2.5.4.3
Raise to the power of .
Step 2.3.2.5.4.4
Use the power rule to combine exponents.
Step 2.3.2.5.4.5
Add and .
Step 2.3.2.5.4.6
Rewrite as .
Step 2.3.2.5.4.6.1
Use to rewrite as .
Step 2.3.2.5.4.6.2
Apply the power rule and multiply exponents, .
Step 2.3.2.5.4.6.3
Combine and .
Step 2.3.2.5.4.6.4
Cancel the common factor of .
Step 2.3.2.5.4.6.4.1
Cancel the common factor.
Step 2.3.2.5.4.6.4.2
Rewrite the expression.
Step 2.3.2.5.4.6.5
Evaluate the exponent.
Step 2.3.2.6
The complete solution is the result of both the positive and negative portions of the solution.
Step 2.3.2.6.1
First, use the positive value of the to find the first solution.
Step 2.3.2.6.2
Next, use the negative value of the to find the second solution.
Step 2.3.2.6.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 2.3.2.7
Set up each of the solutions to solve for .
Step 2.3.2.8
Solve for in .
Step 2.3.2.8.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2.3.2.8.2
Simplify the right side.
Step 2.3.2.8.2.1
The exact value of is .
Step 2.3.2.8.3
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.
Step 2.3.2.8.4
Subtract from .
Step 2.3.2.8.5
Find the period of .
Step 2.3.2.8.5.1
The period of the function can be calculated using .
Step 2.3.2.8.5.2
Replace with in the formula for period.
Step 2.3.2.8.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.3.2.8.5.4
Divide by .
Step 2.3.2.8.6
The period of the function is so values will repeat every degrees in both directions.
, for any integer
, for any integer
Step 2.3.2.9
Solve for in .
Step 2.3.2.9.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2.3.2.9.2
Simplify the right side.
Step 2.3.2.9.2.1
The exact value of is .
Step 2.3.2.9.3
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.
Step 2.3.2.9.4
Simplify the expression to find the second solution.
Step 2.3.2.9.4.1
Subtract from .
Step 2.3.2.9.4.2
The resulting angle of is positive, less than , and coterminal with .
Step 2.3.2.9.5
Find the period of .
Step 2.3.2.9.5.1
The period of the function can be calculated using .
Step 2.3.2.9.5.2
Replace with in the formula for period.
Step 2.3.2.9.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.3.2.9.5.4
Divide by .
Step 2.3.2.9.6
Add to every negative angle to get positive angles.
Step 2.3.2.9.6.1
Add to to find the positive angle.
Step 2.3.2.9.6.2
Subtract from .
Step 2.3.2.9.6.3
List the new angles.
Step 2.3.2.9.7
The period of the function is so values will repeat every degrees in both directions.
, for any integer
, for any integer
Step 2.3.2.10
List all of the solutions.
, for any integer
Step 2.3.2.11
Consolidate the answers.
, for any integer
, for any integer
, for any integer
Step 2.4
The final solution is all the values that make true.
, for any integer
Step 2.5
Consolidate the answers.
Step 2.5.1
Consolidate and to .
, for any integer
Step 2.5.2
Consolidate the answers.
, for any integer
, for any integer
, for any integer
Step 3
The domain is all values of that make the expression defined.
Set-Builder Notation:
, for any integer
Step 4