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Trigonometry Examples
Step 1
Interchange the variables.
Step 2
Step 2.1
Rewrite the equation as .
Step 2.2
Simplify both sides of the equation.
Step 2.2.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 2.2.2
Simplify the denominator.
Step 2.2.2.1
Rewrite as .
Step 2.2.2.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.2.2.3
Simplify.
Step 2.2.2.3.1
Write as a fraction with a common denominator.
Step 2.2.2.3.2
Combine the numerators over the common denominator.
Step 2.2.2.3.3
Multiply by .
Step 2.2.2.3.4
Combine and simplify the denominator.
Step 2.2.2.3.4.1
Multiply by .
Step 2.2.2.3.4.2
Raise to the power of .
Step 2.2.2.3.4.3
Raise to the power of .
Step 2.2.2.3.4.4
Use the power rule to combine exponents.
Step 2.2.2.3.4.5
Add and .
Step 2.2.2.3.4.6
Rewrite as .
Step 2.2.2.3.4.6.1
Use to rewrite as .
Step 2.2.2.3.4.6.2
Apply the power rule and multiply exponents, .
Step 2.2.2.3.4.6.3
Combine and .
Step 2.2.2.3.4.6.4
Cancel the common factor of .
Step 2.2.2.3.4.6.4.1
Cancel the common factor.
Step 2.2.2.3.4.6.4.2
Rewrite the expression.
Step 2.2.2.3.4.6.5
Simplify.
Step 2.2.2.3.5
Write as a fraction with a common denominator.
Step 2.2.2.3.6
Combine the numerators over the common denominator.
Step 2.2.2.4
Multiply by .
Step 2.2.2.5
Combine and simplify the denominator.
Step 2.2.2.5.1
Multiply by .
Step 2.2.2.5.2
Raise to the power of .
Step 2.2.2.5.3
Raise to the power of .
Step 2.2.2.5.4
Use the power rule to combine exponents.
Step 2.2.2.5.5
Add and .
Step 2.2.2.5.6
Rewrite as .
Step 2.2.2.5.6.1
Use to rewrite as .
Step 2.2.2.5.6.2
Apply the power rule and multiply exponents, .
Step 2.2.2.5.6.3
Combine and .
Step 2.2.2.5.6.4
Cancel the common factor of .
Step 2.2.2.5.6.4.1
Cancel the common factor.
Step 2.2.2.5.6.4.2
Rewrite the expression.
Step 2.2.2.5.6.5
Simplify.
Step 2.2.2.6
Apply the distributive property.
Step 2.2.2.7
Multiply .
Step 2.2.2.7.1
Raise to the power of .
Step 2.2.2.7.2
Raise to the power of .
Step 2.2.2.7.3
Use the power rule to combine exponents.
Step 2.2.2.7.4
Add and .
Step 2.2.2.8
Rewrite as .
Step 2.2.2.8.1
Use to rewrite as .
Step 2.2.2.8.2
Apply the power rule and multiply exponents, .
Step 2.2.2.8.3
Combine and .
Step 2.2.2.8.4
Cancel the common factor of .
Step 2.2.2.8.4.1
Cancel the common factor.
Step 2.2.2.8.4.2
Rewrite the expression.
Step 2.2.2.8.5
Simplify.
Step 2.2.2.9
Multiply by .
Step 2.2.2.10
Simplify the denominator.
Step 2.2.2.10.1
Raise to the power of .
Step 2.2.2.10.2
Raise to the power of .
Step 2.2.2.10.3
Use the power rule to combine exponents.
Step 2.2.2.10.4
Add and .
Step 2.2.2.11
Rewrite as .
Step 2.2.2.11.1
Factor the perfect power out of .
Step 2.2.2.11.2
Factor the perfect power out of .
Step 2.2.2.11.3
Rearrange the fraction .
Step 2.2.2.12
Pull terms out from under the radical.
Step 2.2.2.13
Combine and .
Step 2.2.3
Multiply the numerator by the reciprocal of the denominator.
Step 2.2.4
Multiply by .
Step 2.2.5
Multiply by .
Step 2.2.6
Combine and simplify the denominator.
Step 2.2.6.1
Multiply by .
Step 2.2.6.2
Raise to the power of .
Step 2.2.6.3
Raise to the power of .
Step 2.2.6.4
Use the power rule to combine exponents.
Step 2.2.6.5
Add and .
Step 2.2.6.6
Rewrite as .
Step 2.2.6.6.1
Use to rewrite as .
Step 2.2.6.6.2
Apply the power rule and multiply exponents, .
Step 2.2.6.6.3
Combine and .
Step 2.2.6.6.4
Cancel the common factor of .
Step 2.2.6.6.4.1
Cancel the common factor.
Step 2.2.6.6.4.2
Rewrite the expression.
Step 2.2.6.6.5
Simplify.
Step 2.2.7
Multiply by .
Step 2.2.8
Multiply by .
Step 2.2.9
Move .
Step 2.2.10
Expand the denominator using the FOIL method.
Step 2.2.11
Simplify.
Step 2.2.11.1
Subtract from .
Step 2.2.11.2
Add and .
Step 2.2.11.3
Subtract from .
Step 2.2.11.4
Add and .
Step 2.2.12
Cancel the common factor of .
Step 2.2.12.1
Cancel the common factor.
Step 2.2.12.2
Rewrite the expression.
Step 2.2.13
Factor out of .
Step 2.2.13.1
Factor out of .
Step 2.2.13.2
Factor out of .
Step 2.2.13.3
Factor out of .
Step 2.2.14
Cancel the common factor of .
Step 2.2.14.1
Cancel the common factor.
Step 2.2.14.2
Rewrite the expression.
Step 2.3
Simplify the left side.
Step 2.3.1
Use to rewrite as .
Step 2.3.2
Use to rewrite as .
Step 2.3.3
Use to rewrite as .
Step 2.4
Multiply both sides by .
Step 2.5
Simplify.
Step 2.5.1
Simplify the left side.
Step 2.5.1.1
Simplify .
Step 2.5.1.1.1
Simplify the numerator.
Step 2.5.1.1.1.1
Expand by multiplying each term in the first expression by each term in the second expression.
Step 2.5.1.1.1.2
Combine the opposite terms in .
Step 2.5.1.1.1.2.1
Reorder the factors in the terms and .
Step 2.5.1.1.1.2.2
Add and .
Step 2.5.1.1.1.2.3
Add and .
Step 2.5.1.1.1.3
Simplify each term.
Step 2.5.1.1.1.3.1
Multiply by by adding the exponents.
Step 2.5.1.1.1.3.1.1
Use the power rule to combine exponents.
Step 2.5.1.1.1.3.1.2
Add and .
Step 2.5.1.1.1.3.2
Move to the left of .
Step 2.5.1.1.1.3.3
Multiply by .
Step 2.5.1.1.1.3.4
Multiply by .
Step 2.5.1.1.1.3.5
Move to the left of .
Step 2.5.1.1.1.3.6
Rewrite using the commutative property of multiplication.
Step 2.5.1.1.1.3.7
Multiply by by adding the exponents.
Step 2.5.1.1.1.3.7.1
Move .
Step 2.5.1.1.1.3.7.2
Multiply by .
Step 2.5.1.1.1.3.8
Multiply by by adding the exponents.
Step 2.5.1.1.1.3.8.1
Move .
Step 2.5.1.1.1.3.8.2
Use the power rule to combine exponents.
Step 2.5.1.1.1.3.8.3
Combine the numerators over the common denominator.
Step 2.5.1.1.1.3.8.4
Add and .
Step 2.5.1.1.1.3.8.5
Divide by .
Step 2.5.1.1.1.3.9
Simplify .
Step 2.5.1.1.1.3.10
Apply the distributive property.
Step 2.5.1.1.1.3.11
Multiply by by adding the exponents.
Step 2.5.1.1.1.3.11.1
Move .
Step 2.5.1.1.1.3.11.2
Use the power rule to combine exponents.
Step 2.5.1.1.1.3.11.3
Add and .
Step 2.5.1.1.1.3.12
Multiply by .
Step 2.5.1.1.1.4
Combine the opposite terms in .
Step 2.5.1.1.1.4.1
Add and .
Step 2.5.1.1.1.4.2
Add and .
Step 2.5.1.1.1.4.3
Subtract from .
Step 2.5.1.1.1.4.4
Add and .
Step 2.5.1.1.1.4.5
Subtract from .
Step 2.5.1.1.1.4.6
Add and .
Step 2.5.1.1.2
Cancel the common factor of .
Step 2.5.1.1.2.1
Cancel the common factor.
Step 2.5.1.1.2.2
Rewrite the expression.
Step 2.5.2
Simplify the right side.
Step 2.5.2.1
Move to the left of .
Step 2.6
Solve for .
Step 2.6.1
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 2.6.2
Simplify the exponent.
Step 2.6.2.1
Simplify the left side.
Step 2.6.2.1.1
Simplify .
Step 2.6.2.1.1.1
Multiply the exponents in .
Step 2.6.2.1.1.1.1
Apply the power rule and multiply exponents, .
Step 2.6.2.1.1.1.2
Cancel the common factor of .
Step 2.6.2.1.1.1.2.1
Cancel the common factor.
Step 2.6.2.1.1.1.2.2
Rewrite the expression.
Step 2.6.2.1.1.2
Simplify.
Step 2.6.2.2
Simplify the right side.
Step 2.6.2.2.1
Simplify .
Step 2.6.2.2.1.1
Apply the product rule to .
Step 2.6.2.2.1.2
Raise to the power of .
Step 2.6.3
Solve for .
Step 2.6.3.1
Subtract from both sides of the equation.
Step 2.6.3.2
Divide each term in by and simplify.
Step 2.6.3.2.1
Divide each term in by .
Step 2.6.3.2.2
Simplify the left side.
Step 2.6.3.2.2.1
Cancel the common factor of .
Step 2.6.3.2.2.1.1
Cancel the common factor.
Step 2.6.3.2.2.1.2
Divide by .
Step 2.6.3.2.3
Simplify the right side.
Step 2.6.3.2.3.1
Simplify each term.
Step 2.6.3.2.3.1.1
Cancel the common factor of and .
Step 2.6.3.2.3.1.1.1
Factor out of .
Step 2.6.3.2.3.1.1.2
Cancel the common factors.
Step 2.6.3.2.3.1.1.2.1
Factor out of .
Step 2.6.3.2.3.1.1.2.2
Cancel the common factor.
Step 2.6.3.2.3.1.1.2.3
Rewrite the expression.
Step 2.6.3.2.3.1.1.2.4
Divide by .
Step 2.6.3.2.3.1.2
Divide by .
Step 2.6.3.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.6.3.4
Simplify .
Step 2.6.3.4.1
Factor out of .
Step 2.6.3.4.1.1
Factor out of .
Step 2.6.3.4.1.2
Factor out of .
Step 2.6.3.4.1.3
Factor out of .
Step 2.6.3.4.2
Rewrite as .
Step 2.6.3.4.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.6.3.4.4
Rewrite as .
Step 2.6.3.4.4.1
Rewrite as .
Step 2.6.3.4.4.2
Rewrite as .
Step 2.6.3.4.4.3
Add parentheses.
Step 2.6.3.4.5
Pull terms out from under the radical.
Step 2.6.3.4.6
One to any power is one.
Step 2.6.3.5
The complete solution is the result of both the positive and negative portions of the solution.
Step 2.6.3.5.1
First, use the positive value of the to find the first solution.
Step 2.6.3.5.2
Next, use the negative value of the to find the second solution.
Step 2.6.3.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 3
Replace with to show the final answer.
Step 4
Step 4.1
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 4.2
Find the domain of .
Step 4.2.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 4.2.2
Solve for .
Step 4.2.2.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4.2.2.2
Set equal to and solve for .
Step 4.2.2.2.1
Set equal to .
Step 4.2.2.2.2
Subtract from both sides of the equation.
Step 4.2.2.3
Set equal to and solve for .
Step 4.2.2.3.1
Set equal to .
Step 4.2.2.3.2
Add to both sides of the equation.
Step 4.2.2.4
The final solution is all the values that make true.
Step 4.2.2.5
Use each root to create test intervals.
Step 4.2.2.6
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
Step 4.2.2.6.1
Test a value on the interval to see if it makes the inequality true.
Step 4.2.2.6.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 4.2.2.6.1.2
Replace with in the original inequality.
Step 4.2.2.6.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 4.2.2.6.2
Test a value on the interval to see if it makes the inequality true.
Step 4.2.2.6.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 4.2.2.6.2.2
Replace with in the original inequality.
Step 4.2.2.6.2.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 4.2.2.6.3
Test a value on the interval to see if it makes the inequality true.
Step 4.2.2.6.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 4.2.2.6.3.2
Replace with in the original inequality.
Step 4.2.2.6.3.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 4.2.2.6.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 4.2.2.7
The solution consists of all of the true intervals.
or
or
Step 4.2.3
The domain is all values of that make the expression defined.
Step 4.3
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 5