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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
Find the LCD of the terms in the equation.
Step 3.2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 3.2.2
The LCM of one and any expression is the expression.
Step 3.3
Multiply each term in by to eliminate the fractions.
Step 3.3.1
Multiply each term in by .
Step 3.3.2
Simplify the left side.
Step 3.3.2.1
Cancel the common factor of .
Step 3.3.2.1.1
Cancel the common factor.
Step 3.3.2.1.2
Rewrite the expression.
Step 3.4
Solve the equation.
Step 3.4.1
Rewrite the equation as .
Step 3.4.2
Divide each term in by and simplify.
Step 3.4.2.1
Divide each term in by .
Step 3.4.2.2
Simplify the left side.
Step 3.4.2.2.1
Cancel the common factor of .
Step 3.4.2.2.1.1
Cancel the common factor.
Step 3.4.2.2.1.2
Divide by .
Step 3.4.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.4.4
Simplify .
Step 3.4.4.1
Rewrite as .
Step 3.4.4.2
Multiply by .
Step 3.4.4.3
Combine and simplify the denominator.
Step 3.4.4.3.1
Multiply by .
Step 3.4.4.3.2
Raise to the power of .
Step 3.4.4.3.3
Raise to the power of .
Step 3.4.4.3.4
Use the power rule to combine exponents.
Step 3.4.4.3.5
Add and .
Step 3.4.4.3.6
Rewrite as .
Step 3.4.4.3.6.1
Use to rewrite as .
Step 3.4.4.3.6.2
Apply the power rule and multiply exponents, .
Step 3.4.4.3.6.3
Combine and .
Step 3.4.4.3.6.4
Cancel the common factor of .
Step 3.4.4.3.6.4.1
Cancel the common factor.
Step 3.4.4.3.6.4.2
Rewrite the expression.
Step 3.4.4.3.6.5
Simplify.
Step 3.4.4.4
Combine using the product rule for radicals.
Step 3.4.5
The complete solution is the result of both the positive and negative portions of the solution.
Step 3.4.5.1
First, use the positive value of the to find the first solution.
Step 3.4.5.2
Next, use the negative value of the to find the second solution.
Step 3.4.5.3
The complete solution is the result of both the positive and negative portions of the solution.
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
Divide each term in by and simplify.
Step 5.3.2.1
Divide each term in by .
Step 5.3.2.2
Simplify the left side.
Step 5.3.2.2.1
Cancel the common factor of .
Step 5.3.2.2.1.1
Cancel the common factor.
Step 5.3.2.2.1.2
Divide by .
Step 5.3.2.3
Simplify the right side.
Step 5.3.2.3.1
Divide by .
Step 5.3.3
Set the denominator in equal to to find where the expression is undefined.
Step 5.3.4
The domain is all values of that make the expression defined.
Step 5.4
Find the domain of .
Step 5.4.1
Set the denominator in equal to to find where the expression is undefined.
Step 5.4.2
Solve for .
Step 5.4.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.4.2.2
Simplify .
Step 5.4.2.2.1
Rewrite as .
Step 5.4.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 5.4.2.2.3
Plus or minus is .
Step 5.4.3
The domain is all values of that make the expression defined.
Step 5.5
Since the domain of is the range of and the range of is the domain of , then is the inverse of .
Step 6