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Trigonometry Examples
Step 1
Use the quadratic formula to find the solutions.
Step 2
Substitute the values , , and into the quadratic formula and solve for .
Step 3
Step 3.1
Simplify the numerator.
Step 3.1.1
Raise to the power of .
Step 3.1.2
Multiply by .
Step 3.1.3
Apply the distributive property.
Step 3.1.4
Simplify.
Step 3.1.4.1
Multiply by .
Step 3.1.4.2
Multiply by .
Step 3.1.4.3
Multiply by .
Step 3.1.5
Add and .
Step 3.1.6
Rewrite in a factored form.
Step 3.1.6.1
Factor out of .
Step 3.1.6.1.1
Factor out of .
Step 3.1.6.1.2
Factor out of .
Step 3.1.6.1.3
Factor out of .
Step 3.1.6.1.4
Factor out of .
Step 3.1.6.1.5
Factor out of .
Step 3.1.6.2
Factor by grouping.
Step 3.1.6.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 3.1.6.2.1.1
Factor out of .
Step 3.1.6.2.1.2
Rewrite as plus
Step 3.1.6.2.1.3
Apply the distributive property.
Step 3.1.6.2.2
Factor out the greatest common factor from each group.
Step 3.1.6.2.2.1
Group the first two terms and the last two terms.
Step 3.1.6.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 3.1.6.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 3.1.7
Rewrite as .
Step 3.1.7.1
Factor out of .
Step 3.1.7.2
Rewrite as .
Step 3.1.7.3
Rewrite as .
Step 3.1.7.4
Add parentheses.
Step 3.1.7.5
Add parentheses.
Step 3.1.8
Pull terms out from under the radical.
Step 3.1.9
Raise to the power of .
Step 3.2
Multiply by .
Step 3.3
Simplify .
Step 4
Step 4.1
Simplify the numerator.
Step 4.1.1
Raise to the power of .
Step 4.1.2
Multiply by .
Step 4.1.3
Apply the distributive property.
Step 4.1.4
Simplify.
Step 4.1.4.1
Multiply by .
Step 4.1.4.2
Multiply by .
Step 4.1.4.3
Multiply by .
Step 4.1.5
Add and .
Step 4.1.6
Rewrite in a factored form.
Step 4.1.6.1
Factor out of .
Step 4.1.6.1.1
Factor out of .
Step 4.1.6.1.2
Factor out of .
Step 4.1.6.1.3
Factor out of .
Step 4.1.6.1.4
Factor out of .
Step 4.1.6.1.5
Factor out of .
Step 4.1.6.2
Factor by grouping.
Step 4.1.6.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 4.1.6.2.1.1
Factor out of .
Step 4.1.6.2.1.2
Rewrite as plus
Step 4.1.6.2.1.3
Apply the distributive property.
Step 4.1.6.2.2
Factor out the greatest common factor from each group.
Step 4.1.6.2.2.1
Group the first two terms and the last two terms.
Step 4.1.6.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 4.1.6.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 4.1.7
Rewrite as .
Step 4.1.7.1
Factor out of .
Step 4.1.7.2
Rewrite as .
Step 4.1.7.3
Rewrite as .
Step 4.1.7.4
Add parentheses.
Step 4.1.7.5
Add parentheses.
Step 4.1.8
Pull terms out from under the radical.
Step 4.1.9
Raise to the power of .
Step 4.2
Multiply by .
Step 4.3
Simplify .
Step 4.4
Change the to .
Step 5
Step 5.1
Simplify the numerator.
Step 5.1.1
Raise to the power of .
Step 5.1.2
Multiply by .
Step 5.1.3
Apply the distributive property.
Step 5.1.4
Simplify.
Step 5.1.4.1
Multiply by .
Step 5.1.4.2
Multiply by .
Step 5.1.4.3
Multiply by .
Step 5.1.5
Add and .
Step 5.1.6
Rewrite in a factored form.
Step 5.1.6.1
Factor out of .
Step 5.1.6.1.1
Factor out of .
Step 5.1.6.1.2
Factor out of .
Step 5.1.6.1.3
Factor out of .
Step 5.1.6.1.4
Factor out of .
Step 5.1.6.1.5
Factor out of .
Step 5.1.6.2
Factor by grouping.
Step 5.1.6.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 5.1.6.2.1.1
Factor out of .
Step 5.1.6.2.1.2
Rewrite as plus
Step 5.1.6.2.1.3
Apply the distributive property.
Step 5.1.6.2.2
Factor out the greatest common factor from each group.
Step 5.1.6.2.2.1
Group the first two terms and the last two terms.
Step 5.1.6.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 5.1.6.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 5.1.7
Rewrite as .
Step 5.1.7.1
Factor out of .
Step 5.1.7.2
Rewrite as .
Step 5.1.7.3
Rewrite as .
Step 5.1.7.4
Add parentheses.
Step 5.1.7.5
Add parentheses.
Step 5.1.8
Pull terms out from under the radical.
Step 5.1.9
Raise to the power of .
Step 5.2
Multiply by .
Step 5.3
Simplify .
Step 5.4
Change the to .
Step 6
The final answer is the combination of both solutions.
Step 7
Interchange the variables. Create an equation for each expression.
Step 8
Step 8.1
Rewrite the equation as .
Step 8.2
Multiply both sides by .
Step 8.3
Simplify.
Step 8.3.1
Simplify the left side.
Step 8.3.1.1
Simplify .
Step 8.3.1.1.1
Cancel the common factor of .
Step 8.3.1.1.1.1
Cancel the common factor.
Step 8.3.1.1.1.2
Rewrite the expression.
Step 8.3.1.1.2
Reorder and .
Step 8.3.2
Simplify the right side.
Step 8.3.2.1
Move to the left of .
Step 8.4
Solve for .
Step 8.4.1
Subtract from both sides of the equation.
Step 8.4.2
To remove the radical on the left side of the equation, square both sides of the equation.
Step 8.4.3
Simplify each side of the equation.
Step 8.4.3.1
Use to rewrite as .
Step 8.4.3.2
Simplify the left side.
Step 8.4.3.2.1
Simplify .
Step 8.4.3.2.1.1
Simplify by multiplying through.
Step 8.4.3.2.1.1.1
Apply the distributive property.
Step 8.4.3.2.1.1.2
Multiply.
Step 8.4.3.2.1.1.2.1
Multiply by .
Step 8.4.3.2.1.1.2.2
Multiply by .
Step 8.4.3.2.1.2
Expand using the FOIL Method.
Step 8.4.3.2.1.2.1
Apply the distributive property.
Step 8.4.3.2.1.2.2
Apply the distributive property.
Step 8.4.3.2.1.2.3
Apply the distributive property.
Step 8.4.3.2.1.3
Simplify and combine like terms.
Step 8.4.3.2.1.3.1
Simplify each term.
Step 8.4.3.2.1.3.1.1
Multiply by by adding the exponents.
Step 8.4.3.2.1.3.1.1.1
Move .
Step 8.4.3.2.1.3.1.1.2
Multiply by .
Step 8.4.3.2.1.3.1.2
Multiply by .
Step 8.4.3.2.1.3.1.3
Multiply by .
Step 8.4.3.2.1.3.2
Add and .
Step 8.4.3.2.1.4
Apply the product rule to .
Step 8.4.3.2.1.5
Raise to the power of .
Step 8.4.3.2.1.6
Multiply the exponents in .
Step 8.4.3.2.1.6.1
Apply the power rule and multiply exponents, .
Step 8.4.3.2.1.6.2
Cancel the common factor of .
Step 8.4.3.2.1.6.2.1
Cancel the common factor.
Step 8.4.3.2.1.6.2.2
Rewrite the expression.
Step 8.4.3.2.1.7
Simplify.
Step 8.4.3.2.1.8
Apply the distributive property.
Step 8.4.3.2.1.9
Simplify.
Step 8.4.3.2.1.9.1
Multiply by .
Step 8.4.3.2.1.9.2
Multiply by .
Step 8.4.3.2.1.9.3
Multiply by .
Step 8.4.3.3
Simplify the right side.
Step 8.4.3.3.1
Simplify .
Step 8.4.3.3.1.1
Rewrite as .
Step 8.4.3.3.1.2
Expand using the FOIL Method.
Step 8.4.3.3.1.2.1
Apply the distributive property.
Step 8.4.3.3.1.2.2
Apply the distributive property.
Step 8.4.3.3.1.2.3
Apply the distributive property.
Step 8.4.3.3.1.3
Simplify and combine like terms.
Step 8.4.3.3.1.3.1
Simplify each term.
Step 8.4.3.3.1.3.1.1
Rewrite using the commutative property of multiplication.
Step 8.4.3.3.1.3.1.2
Multiply by by adding the exponents.
Step 8.4.3.3.1.3.1.2.1
Move .
Step 8.4.3.3.1.3.1.2.2
Multiply by .
Step 8.4.3.3.1.3.1.3
Multiply by .
Step 8.4.3.3.1.3.1.4
Multiply by .
Step 8.4.3.3.1.3.1.5
Multiply by .
Step 8.4.3.3.1.3.1.6
Multiply by .
Step 8.4.3.3.1.3.2
Subtract from .
Step 8.4.4
Solve for .
Step 8.4.4.1
Move all terms to the left side of the equation and simplify.
Step 8.4.4.1.1
Move all the expressions to the left side of the equation.
Step 8.4.4.1.1.1
Subtract from both sides of the equation.
Step 8.4.4.1.1.2
Add to both sides of the equation.
Step 8.4.4.1.1.3
Subtract from both sides of the equation.
Step 8.4.4.1.2
Subtract from .
Step 8.4.4.2
Use the quadratic formula to find the solutions.
Step 8.4.4.3
Substitute the values , , and into the quadratic formula and solve for .
Step 8.4.4.4
Simplify.
Step 8.4.4.4.1
Simplify the numerator.
Step 8.4.4.4.1.1
Raise to the power of .
Step 8.4.4.4.1.2
Multiply by .
Step 8.4.4.4.1.3
Apply the distributive property.
Step 8.4.4.4.1.4
Simplify.
Step 8.4.4.4.1.4.1
Multiply by .
Step 8.4.4.4.1.4.2
Multiply by .
Step 8.4.4.4.1.4.3
Multiply by .
Step 8.4.4.4.1.5
Add and .
Step 8.4.4.4.1.6
Factor out of .
Step 8.4.4.4.1.6.1
Factor out of .
Step 8.4.4.4.1.6.2
Factor out of .
Step 8.4.4.4.1.6.3
Factor out of .
Step 8.4.4.4.1.6.4
Factor out of .
Step 8.4.4.4.1.6.5
Factor out of .
Step 8.4.4.4.1.7
Rewrite as .
Step 8.4.4.4.1.7.1
Factor out of .
Step 8.4.4.4.1.7.2
Rewrite as .
Step 8.4.4.4.1.7.3
Add parentheses.
Step 8.4.4.4.1.8
Pull terms out from under the radical.
Step 8.4.4.4.2
Multiply by .
Step 8.4.4.4.3
Simplify .
Step 8.4.4.4.4
Move the negative in front of the fraction.
Step 8.4.4.5
Simplify the expression to solve for the portion of the .
Step 8.4.4.5.1
Simplify the numerator.
Step 8.4.4.5.1.1
Raise to the power of .
Step 8.4.4.5.1.2
Multiply by .
Step 8.4.4.5.1.3
Apply the distributive property.
Step 8.4.4.5.1.4
Simplify.
Step 8.4.4.5.1.4.1
Multiply by .
Step 8.4.4.5.1.4.2
Multiply by .
Step 8.4.4.5.1.4.3
Multiply by .
Step 8.4.4.5.1.5
Add and .
Step 8.4.4.5.1.6
Factor out of .
Step 8.4.4.5.1.6.1
Factor out of .
Step 8.4.4.5.1.6.2
Factor out of .
Step 8.4.4.5.1.6.3
Factor out of .
Step 8.4.4.5.1.6.4
Factor out of .
Step 8.4.4.5.1.6.5
Factor out of .
Step 8.4.4.5.1.7
Rewrite as .
Step 8.4.4.5.1.7.1
Factor out of .
Step 8.4.4.5.1.7.2
Rewrite as .
Step 8.4.4.5.1.7.3
Add parentheses.
Step 8.4.4.5.1.8
Pull terms out from under the radical.
Step 8.4.4.5.2
Multiply by .
Step 8.4.4.5.3
Simplify .
Step 8.4.4.5.4
Move the negative in front of the fraction.
Step 8.4.4.5.5
Change the to .
Step 8.4.4.6
Simplify the expression to solve for the portion of the .
Step 8.4.4.6.1
Simplify the numerator.
Step 8.4.4.6.1.1
Raise to the power of .
Step 8.4.4.6.1.2
Multiply by .
Step 8.4.4.6.1.3
Apply the distributive property.
Step 8.4.4.6.1.4
Simplify.
Step 8.4.4.6.1.4.1
Multiply by .
Step 8.4.4.6.1.4.2
Multiply by .
Step 8.4.4.6.1.4.3
Multiply by .
Step 8.4.4.6.1.5
Add and .
Step 8.4.4.6.1.6
Factor out of .
Step 8.4.4.6.1.6.1
Factor out of .
Step 8.4.4.6.1.6.2
Factor out of .
Step 8.4.4.6.1.6.3
Factor out of .
Step 8.4.4.6.1.6.4
Factor out of .
Step 8.4.4.6.1.6.5
Factor out of .
Step 8.4.4.6.1.7
Rewrite as .
Step 8.4.4.6.1.7.1
Factor out of .
Step 8.4.4.6.1.7.2
Rewrite as .
Step 8.4.4.6.1.7.3
Add parentheses.
Step 8.4.4.6.1.8
Pull terms out from under the radical.
Step 8.4.4.6.2
Multiply by .
Step 8.4.4.6.3
Simplify .
Step 8.4.4.6.4
Move the negative in front of the fraction.
Step 8.4.4.6.5
Change the to .
Step 8.4.4.7
The final answer is the combination of both solutions.
Step 9
Replace with to show the final answer.
Step 10
Step 10.1
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 10.2
Find the range of .
Step 10.2.1
Find the range of .
Step 10.2.1.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 10.2.2
Find the range of .
Step 10.2.2.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 10.2.3
Find the union of .
Step 10.2.3.1
The union consists of all of the elements that are contained in each interval.
Step 10.3
Find the domain of .
Step 10.3.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 10.3.2
Solve for .
Step 10.3.2.1
Divide each term in by and simplify.
Step 10.3.2.1.1
Divide each term in by .
Step 10.3.2.1.2
Simplify the left side.
Step 10.3.2.1.2.1
Cancel the common factor of .
Step 10.3.2.1.2.1.1
Cancel the common factor.
Step 10.3.2.1.2.1.2
Divide by .
Step 10.3.2.1.3
Simplify the right side.
Step 10.3.2.1.3.1
Divide by .
Step 10.3.2.2
Convert the inequality to an equation.
Step 10.3.2.3
Use the quadratic formula to find the solutions.
Step 10.3.2.4
Substitute the values , , and into the quadratic formula and solve for .
Step 10.3.2.5
Simplify.
Step 10.3.2.5.1
Simplify the numerator.
Step 10.3.2.5.1.1
Raise to the power of .
Step 10.3.2.5.1.2
Multiply .
Step 10.3.2.5.1.2.1
Multiply by .
Step 10.3.2.5.1.2.2
Multiply by .
Step 10.3.2.5.1.3
Add and .
Step 10.3.2.5.1.4
Rewrite as .
Step 10.3.2.5.1.4.1
Factor out of .
Step 10.3.2.5.1.4.2
Rewrite as .
Step 10.3.2.5.1.5
Pull terms out from under the radical.
Step 10.3.2.5.2
Multiply by .
Step 10.3.2.5.3
Simplify .
Step 10.3.2.6
Simplify the expression to solve for the portion of the .
Step 10.3.2.6.1
Simplify the numerator.
Step 10.3.2.6.1.1
Raise to the power of .
Step 10.3.2.6.1.2
Multiply .
Step 10.3.2.6.1.2.1
Multiply by .
Step 10.3.2.6.1.2.2
Multiply by .
Step 10.3.2.6.1.3
Add and .
Step 10.3.2.6.1.4
Rewrite as .
Step 10.3.2.6.1.4.1
Factor out of .
Step 10.3.2.6.1.4.2
Rewrite as .
Step 10.3.2.6.1.5
Pull terms out from under the radical.
Step 10.3.2.6.2
Multiply by .
Step 10.3.2.6.3
Simplify .
Step 10.3.2.6.4
Change the to .
Step 10.3.2.7
Simplify the expression to solve for the portion of the .
Step 10.3.2.7.1
Simplify the numerator.
Step 10.3.2.7.1.1
Raise to the power of .
Step 10.3.2.7.1.2
Multiply .
Step 10.3.2.7.1.2.1
Multiply by .
Step 10.3.2.7.1.2.2
Multiply by .
Step 10.3.2.7.1.3
Add and .
Step 10.3.2.7.1.4
Rewrite as .
Step 10.3.2.7.1.4.1
Factor out of .
Step 10.3.2.7.1.4.2
Rewrite as .
Step 10.3.2.7.1.5
Pull terms out from under the radical.
Step 10.3.2.7.2
Multiply by .
Step 10.3.2.7.3
Simplify .
Step 10.3.2.7.4
Change the to .
Step 10.3.2.8
Consolidate the solutions.
Step 10.3.2.9
Use each root to create test intervals.
Step 10.3.2.10
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
Step 10.3.2.10.1
Test a value on the interval to see if it makes the inequality true.
Step 10.3.2.10.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 10.3.2.10.1.2
Replace with in the original inequality.
Step 10.3.2.10.1.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 10.3.2.10.2
Test a value on the interval to see if it makes the inequality true.
Step 10.3.2.10.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 10.3.2.10.2.2
Replace with in the original inequality.
Step 10.3.2.10.2.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 10.3.2.10.3
Test a value on the interval to see if it makes the inequality true.
Step 10.3.2.10.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 10.3.2.10.3.2
Replace with in the original inequality.
Step 10.3.2.10.3.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 10.3.2.10.4
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
False
True
False
Step 10.3.2.11
The solution consists of all of the true intervals.
Step 10.3.3
The domain is all values of that make the expression defined.
Step 10.4
Find the domain of .
Step 10.4.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 10.4.2
Solve for .
Step 10.4.2.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 10.4.2.2
Set equal to and solve for .
Step 10.4.2.2.1
Set equal to .
Step 10.4.2.2.2
Solve for .
Step 10.4.2.2.2.1
Subtract from both sides of the equation.
Step 10.4.2.2.2.2
Divide each term in by and simplify.
Step 10.4.2.2.2.2.1
Divide each term in by .
Step 10.4.2.2.2.2.2
Simplify the left side.
Step 10.4.2.2.2.2.2.1
Dividing two negative values results in a positive value.
Step 10.4.2.2.2.2.2.2
Divide by .
Step 10.4.2.2.2.2.3
Simplify the right side.
Step 10.4.2.2.2.2.3.1
Divide by .
Step 10.4.2.3
Set equal to and solve for .
Step 10.4.2.3.1
Set equal to .
Step 10.4.2.3.2
Subtract from both sides of the equation.
Step 10.4.2.4
The final solution is all the values that make true.
Step 10.4.2.5
Use each root to create test intervals.
Step 10.4.2.6
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
Step 10.4.2.6.1
Test a value on the interval to see if it makes the inequality true.
Step 10.4.2.6.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 10.4.2.6.1.2
Replace with in the original inequality.
Step 10.4.2.6.1.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 10.4.2.6.2
Test a value on the interval to see if it makes the inequality true.
Step 10.4.2.6.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 10.4.2.6.2.2
Replace with in the original inequality.
Step 10.4.2.6.2.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 10.4.2.6.3
Test a value on the interval to see if it makes the inequality true.
Step 10.4.2.6.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 10.4.2.6.3.2
Replace with in the original inequality.
Step 10.4.2.6.3.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 10.4.2.6.4
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
False
True
False
Step 10.4.2.7
The solution consists of all of the true intervals.
Step 10.4.3
The domain is all values of that make the expression defined.
Step 10.5
Since the domain of is the range of and the range of is the domain of , then is the inverse of .
Step 11