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Trigonometry Examples
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Step 3.1
Rewrite the equation as .
Step 3.2
Factor each term.
Step 3.2.1
Factor using the AC method.
Step 3.2.1.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 3.2.1.2
Write the factored form using these integers.
Step 3.2.2
Factor using the AC method.
Step 3.2.2.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 3.2.2.2
Write the factored form using these integers.
Step 3.3
Find the LCD of the terms in the equation.
Step 3.3.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 3.3.2
The LCM of one and any expression is the expression.
Step 3.4
Multiply each term in by to eliminate the fractions.
Step 3.4.1
Multiply each term in by .
Step 3.4.2
Simplify the left side.
Step 3.4.2.1
Cancel the common factor of .
Step 3.4.2.1.1
Cancel the common factor.
Step 3.4.2.1.2
Rewrite the expression.
Step 3.4.2.2
Expand using the FOIL Method.
Step 3.4.2.2.1
Apply the distributive property.
Step 3.4.2.2.2
Apply the distributive property.
Step 3.4.2.2.3
Apply the distributive property.
Step 3.4.2.3
Simplify and combine like terms.
Step 3.4.2.3.1
Simplify each term.
Step 3.4.2.3.1.1
Multiply by .
Step 3.4.2.3.1.2
Move to the left of .
Step 3.4.2.3.1.3
Multiply by .
Step 3.4.2.3.2
Subtract from .
Step 3.4.3
Simplify the right side.
Step 3.4.3.1
Expand using the FOIL Method.
Step 3.4.3.1.1
Apply the distributive property.
Step 3.4.3.1.2
Apply the distributive property.
Step 3.4.3.1.3
Apply the distributive property.
Step 3.4.3.2
Simplify and combine like terms.
Step 3.4.3.2.1
Simplify each term.
Step 3.4.3.2.1.1
Multiply by .
Step 3.4.3.2.1.2
Move to the left of .
Step 3.4.3.2.1.3
Multiply by .
Step 3.4.3.2.2
Subtract from .
Step 3.4.3.3
Apply the distributive property.
Step 3.4.3.4
Simplify.
Step 3.4.3.4.1
Rewrite using the commutative property of multiplication.
Step 3.4.3.4.2
Move to the left of .
Step 3.5
Solve the equation.
Step 3.5.1
Move all terms containing to the left side of the equation.
Step 3.5.1.1
Subtract from both sides of the equation.
Step 3.5.1.2
Subtract from both sides of the equation.
Step 3.5.2
Add to both sides of the equation.
Step 3.5.3
Use the quadratic formula to find the solutions.
Step 3.5.4
Substitute the values , , and into the quadratic formula and solve for .
Step 3.5.5
Simplify the numerator.
Step 3.5.5.1
Apply the distributive property.
Step 3.5.5.2
Multiply by .
Step 3.5.5.3
Multiply by .
Step 3.5.5.4
Rewrite as .
Step 3.5.5.5
Expand using the FOIL Method.
Step 3.5.5.5.1
Apply the distributive property.
Step 3.5.5.5.2
Apply the distributive property.
Step 3.5.5.5.3
Apply the distributive property.
Step 3.5.5.6
Simplify and combine like terms.
Step 3.5.5.6.1
Simplify each term.
Step 3.5.5.6.1.1
Multiply by .
Step 3.5.5.6.1.2
Multiply by .
Step 3.5.5.6.1.3
Multiply by .
Step 3.5.5.6.1.4
Rewrite using the commutative property of multiplication.
Step 3.5.5.6.1.5
Multiply by by adding the exponents.
Step 3.5.5.6.1.5.1
Move .
Step 3.5.5.6.1.5.2
Multiply by .
Step 3.5.5.6.1.6
Multiply by .
Step 3.5.5.6.2
Add and .
Step 3.5.5.7
Apply the distributive property.
Step 3.5.5.8
Multiply by .
Step 3.5.5.9
Multiply by .
Step 3.5.5.10
Expand using the FOIL Method.
Step 3.5.5.10.1
Apply the distributive property.
Step 3.5.5.10.2
Apply the distributive property.
Step 3.5.5.10.3
Apply the distributive property.
Step 3.5.5.11
Simplify and combine like terms.
Step 3.5.5.11.1
Simplify each term.
Step 3.5.5.11.1.1
Multiply by .
Step 3.5.5.11.1.2
Multiply by .
Step 3.5.5.11.1.3
Multiply by .
Step 3.5.5.11.1.4
Rewrite using the commutative property of multiplication.
Step 3.5.5.11.1.5
Multiply by by adding the exponents.
Step 3.5.5.11.1.5.1
Move .
Step 3.5.5.11.1.5.2
Multiply by .
Step 3.5.5.11.1.6
Multiply by .
Step 3.5.5.11.2
Subtract from .
Step 3.5.5.12
Add and .
Step 3.5.5.13
Subtract from .
Step 3.5.5.14
Add and .
Step 3.5.5.15
Reorder terms.
Step 3.5.6
Change the to .
Step 3.5.7
Simplify the expression to solve for the portion of the .
Step 3.5.7.1
Simplify the numerator.
Step 3.5.7.1.1
Apply the distributive property.
Step 3.5.7.1.2
Multiply by .
Step 3.5.7.1.3
Multiply by .
Step 3.5.7.1.4
Rewrite as .
Step 3.5.7.1.5
Expand using the FOIL Method.
Step 3.5.7.1.5.1
Apply the distributive property.
Step 3.5.7.1.5.2
Apply the distributive property.
Step 3.5.7.1.5.3
Apply the distributive property.
Step 3.5.7.1.6
Simplify and combine like terms.
Step 3.5.7.1.6.1
Simplify each term.
Step 3.5.7.1.6.1.1
Multiply by .
Step 3.5.7.1.6.1.2
Multiply by .
Step 3.5.7.1.6.1.3
Multiply by .
Step 3.5.7.1.6.1.4
Rewrite using the commutative property of multiplication.
Step 3.5.7.1.6.1.5
Multiply by by adding the exponents.
Step 3.5.7.1.6.1.5.1
Move .
Step 3.5.7.1.6.1.5.2
Multiply by .
Step 3.5.7.1.6.1.6
Multiply by .
Step 3.5.7.1.6.2
Add and .
Step 3.5.7.1.7
Apply the distributive property.
Step 3.5.7.1.8
Multiply by .
Step 3.5.7.1.9
Multiply by .
Step 3.5.7.1.10
Expand using the FOIL Method.
Step 3.5.7.1.10.1
Apply the distributive property.
Step 3.5.7.1.10.2
Apply the distributive property.
Step 3.5.7.1.10.3
Apply the distributive property.
Step 3.5.7.1.11
Simplify and combine like terms.
Step 3.5.7.1.11.1
Simplify each term.
Step 3.5.7.1.11.1.1
Multiply by .
Step 3.5.7.1.11.1.2
Multiply by .
Step 3.5.7.1.11.1.3
Multiply by .
Step 3.5.7.1.11.1.4
Rewrite using the commutative property of multiplication.
Step 3.5.7.1.11.1.5
Multiply by by adding the exponents.
Step 3.5.7.1.11.1.5.1
Move .
Step 3.5.7.1.11.1.5.2
Multiply by .
Step 3.5.7.1.11.1.6
Multiply by .
Step 3.5.7.1.11.2
Subtract from .
Step 3.5.7.1.12
Add and .
Step 3.5.7.1.13
Subtract from .
Step 3.5.7.1.14
Add and .
Step 3.5.7.1.15
Reorder terms.
Step 3.5.7.2
Change the to .
Step 3.5.8
The final answer is the combination of both solutions.
Step 4
Replace with to show the final answer.
Step 5
Step 5.1
The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of and and compare them.
Step 5.2
Find the range of .
Step 5.2.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 5.3
Find the domain of .
Step 5.3.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 5.3.2
Solve for .
Step 5.3.2.1
Convert the inequality to an equation.
Step 5.3.2.2
Use the quadratic formula to find the solutions.
Step 5.3.2.3
Substitute the values , , and into the quadratic formula and solve for .
Step 5.3.2.4
Simplify.
Step 5.3.2.4.1
Simplify the numerator.
Step 5.3.2.4.1.1
Raise to the power of .
Step 5.3.2.4.1.2
Multiply .
Step 5.3.2.4.1.2.1
Multiply by .
Step 5.3.2.4.1.2.2
Multiply by .
Step 5.3.2.4.1.3
Subtract from .
Step 5.3.2.4.1.4
Rewrite as .
Step 5.3.2.4.1.4.1
Factor out of .
Step 5.3.2.4.1.4.2
Rewrite as .
Step 5.3.2.4.1.5
Pull terms out from under the radical.
Step 5.3.2.4.2
Multiply by .
Step 5.3.2.4.3
Simplify .
Step 5.3.2.5
Simplify the expression to solve for the portion of the .
Step 5.3.2.5.1
Simplify the numerator.
Step 5.3.2.5.1.1
Raise to the power of .
Step 5.3.2.5.1.2
Multiply .
Step 5.3.2.5.1.2.1
Multiply by .
Step 5.3.2.5.1.2.2
Multiply by .
Step 5.3.2.5.1.3
Subtract from .
Step 5.3.2.5.1.4
Rewrite as .
Step 5.3.2.5.1.4.1
Factor out of .
Step 5.3.2.5.1.4.2
Rewrite as .
Step 5.3.2.5.1.5
Pull terms out from under the radical.
Step 5.3.2.5.2
Multiply by .
Step 5.3.2.5.3
Simplify .
Step 5.3.2.5.4
Change the to .
Step 5.3.2.6
Simplify the expression to solve for the portion of the .
Step 5.3.2.6.1
Simplify the numerator.
Step 5.3.2.6.1.1
Raise to the power of .
Step 5.3.2.6.1.2
Multiply .
Step 5.3.2.6.1.2.1
Multiply by .
Step 5.3.2.6.1.2.2
Multiply by .
Step 5.3.2.6.1.3
Subtract from .
Step 5.3.2.6.1.4
Rewrite as .
Step 5.3.2.6.1.4.1
Factor out of .
Step 5.3.2.6.1.4.2
Rewrite as .
Step 5.3.2.6.1.5
Pull terms out from under the radical.
Step 5.3.2.6.2
Multiply by .
Step 5.3.2.6.3
Simplify .
Step 5.3.2.6.4
Change the to .
Step 5.3.2.7
Consolidate the solutions.
Step 5.3.2.8
Use each root to create test intervals.
Step 5.3.2.9
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
Step 5.3.2.9.1
Test a value on the interval to see if it makes the inequality true.
Step 5.3.2.9.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.2.9.1.2
Replace with in the original inequality.
Step 5.3.2.9.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 5.3.2.9.2
Test a value on the interval to see if it makes the inequality true.
Step 5.3.2.9.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.2.9.2.2
Replace with in the original inequality.
Step 5.3.2.9.2.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 5.3.2.9.3
Test a value on the interval to see if it makes the inequality true.
Step 5.3.2.9.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.2.9.3.2
Replace with in the original inequality.
Step 5.3.2.9.3.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 5.3.2.9.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 5.3.2.10
The solution consists of all of the true intervals.
or
or
Step 5.3.3
Set the denominator in equal to to find where the expression is undefined.
Step 5.3.4
Solve for .
Step 5.3.4.1
Divide each term in by and simplify.
Step 5.3.4.1.1
Divide each term in by .
Step 5.3.4.1.2
Simplify the left side.
Step 5.3.4.1.2.1
Cancel the common factor of .
Step 5.3.4.1.2.1.1
Cancel the common factor.
Step 5.3.4.1.2.1.2
Divide by .
Step 5.3.4.1.3
Simplify the right side.
Step 5.3.4.1.3.1
Divide by .
Step 5.3.4.2
Subtract from both sides of the equation.
Step 5.3.4.3
Divide each term in by and simplify.
Step 5.3.4.3.1
Divide each term in by .
Step 5.3.4.3.2
Simplify the left side.
Step 5.3.4.3.2.1
Dividing two negative values results in a positive value.
Step 5.3.4.3.2.2
Divide by .
Step 5.3.4.3.3
Simplify the right side.
Step 5.3.4.3.3.1
Divide by .
Step 5.3.5
The domain is all values of that make the expression defined.
Step 5.4
Since the domain of is not equal to the range of , then is not an inverse of .
There is no inverse
There is no inverse
Step 6