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Trigonometry Examples
Step 1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2
Step 2.1
First, use the positive value of the to find the first solution.
Step 2.2
Add to both sides of the equation.
Step 2.3
Next, use the negative value of the to find the second solution.
Step 2.4
Add to both sides of the equation.
Step 2.5
The complete solution is the result of both the positive and negative portions of the solution.
Step 3
Interchange the variables. Create an equation for each expression.
Step 4
Step 4.1
Rewrite the equation as .
Step 4.2
Subtract from both sides of the equation.
Step 4.3
To remove the radical on the left side of the equation, square both sides of the equation.
Step 4.4
Simplify each side of the equation.
Step 4.4.1
Use to rewrite as .
Step 4.4.2
Simplify the left side.
Step 4.4.2.1
Simplify .
Step 4.4.2.1.1
Multiply the exponents in .
Step 4.4.2.1.1.1
Apply the power rule and multiply exponents, .
Step 4.4.2.1.1.2
Cancel the common factor of .
Step 4.4.2.1.1.2.1
Cancel the common factor.
Step 4.4.2.1.1.2.2
Rewrite the expression.
Step 4.4.2.1.2
Apply the distributive property.
Step 4.4.2.1.3
Simplify.
Step 4.4.2.1.3.1
Multiply by .
Step 4.4.2.1.3.2
Simplify.
Step 4.4.3
Simplify the right side.
Step 4.4.3.1
Simplify .
Step 4.4.3.1.1
Rewrite as .
Step 4.4.3.1.2
Expand using the FOIL Method.
Step 4.4.3.1.2.1
Apply the distributive property.
Step 4.4.3.1.2.2
Apply the distributive property.
Step 4.4.3.1.2.3
Apply the distributive property.
Step 4.4.3.1.3
Simplify and combine like terms.
Step 4.4.3.1.3.1
Simplify each term.
Step 4.4.3.1.3.1.1
Multiply by .
Step 4.4.3.1.3.1.2
Move to the left of .
Step 4.4.3.1.3.1.3
Multiply by .
Step 4.4.3.1.3.2
Subtract from .
Step 4.5
Solve for .
Step 4.5.1
Move all terms not containing to the right side of the equation.
Step 4.5.1.1
Subtract from both sides of the equation.
Step 4.5.1.2
Subtract from .
Step 4.5.2
Divide each term in by and simplify.
Step 4.5.2.1
Divide each term in by .
Step 4.5.2.2
Simplify the left side.
Step 4.5.2.2.1
Cancel the common factor of .
Step 4.5.2.2.1.1
Cancel the common factor.
Step 4.5.2.2.1.2
Divide by .
Step 4.5.2.3
Simplify the right side.
Step 4.5.2.3.1
Move the negative in front of the fraction.
Step 5
Replace with to show the final answer.
Step 6
Step 6.1
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 6.2
Find the range of .
Step 6.2.1
Find the range of .
Step 6.2.1.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 6.2.2
Find the range of .
Step 6.2.2.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 6.2.3
Find the union of .
Step 6.2.3.1
The union consists of all of the elements that are contained in each interval.
Step 6.3
Find the domain of .
Step 6.3.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 6.3.2
Solve for .
Step 6.3.2.1
Divide each term in by and simplify.
Step 6.3.2.1.1
Divide each term in by .
Step 6.3.2.1.2
Simplify the left side.
Step 6.3.2.1.2.1
Cancel the common factor of .
Step 6.3.2.1.2.1.1
Cancel the common factor.
Step 6.3.2.1.2.1.2
Divide by .
Step 6.3.2.1.3
Simplify the right side.
Step 6.3.2.1.3.1
Divide by .
Step 6.3.2.2
Subtract from both sides of the inequality.
Step 6.3.3
The domain is all values of that make the expression defined.
Step 6.4
Since the domain of is the range of and the range of is the domain of , then is the inverse of .
Step 7