Trigonometry Examples

Find the Inverse f^-1(81)
Step 1
Interchange the variables.
Step 2
Solve for .
Tap for more steps...
Step 2.1
Rewrite the equation as .
Step 2.2
Divide each term in by and simplify.
Tap for more steps...
Step 2.2.1
Divide each term in by .
Step 2.2.2
Simplify the left side.
Tap for more steps...
Step 2.2.2.1
Move to the denominator using the negative exponent rule .
Step 2.2.2.2
Cancel the common factor of .
Tap for more steps...
Step 2.2.2.2.1
Cancel the common factor.
Step 2.2.2.2.2
Rewrite the expression.
Step 2.3
Find the LCD of the terms in the equation.
Tap for more steps...
Step 2.3.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 2.3.2
Since contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
Step 2.3.3
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
Step 2.3.4
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 2.3.5
The prime factors for are .
Tap for more steps...
Step 2.3.5.1
has factors of and .
Step 2.3.5.2
has factors of and .
Step 2.3.5.3
has factors of and .
Step 2.3.6
Multiply .
Tap for more steps...
Step 2.3.6.1
Multiply by .
Step 2.3.6.2
Multiply by .
Step 2.3.6.3
Multiply by .
Step 2.3.7
The factor for is itself.
occurs time.
Step 2.3.8
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 2.3.9
The LCM for is the numeric part multiplied by the variable part.
Step 2.4
Multiply each term in by to eliminate the fractions.
Tap for more steps...
Step 2.4.1
Multiply each term in by .
Step 2.4.2
Simplify the left side.
Tap for more steps...
Step 2.4.2.1
Rewrite using the commutative property of multiplication.
Step 2.4.2.2
Combine and .
Step 2.4.2.3
Cancel the common factor of .
Tap for more steps...
Step 2.4.2.3.1
Cancel the common factor.
Step 2.4.2.3.2
Rewrite the expression.
Step 2.4.3
Simplify the right side.
Tap for more steps...
Step 2.4.3.1
Rewrite using the commutative property of multiplication.
Step 2.4.3.2
Cancel the common factor of .
Tap for more steps...
Step 2.4.3.2.1
Cancel the common factor.
Step 2.4.3.2.2
Rewrite the expression.
Step 2.5
Solve the equation.
Tap for more steps...
Step 2.5.1
Rewrite the equation as .
Step 2.5.2
Divide each term in by and simplify.
Tap for more steps...
Step 2.5.2.1
Divide each term in by .
Step 2.5.2.2
Simplify the left side.
Tap for more steps...
Step 2.5.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 2.5.2.2.1.1
Cancel the common factor.
Step 2.5.2.2.1.2
Divide by .
Step 3
Replace with to show the final answer.
Step 4
Verify if is the inverse of .
Tap for more steps...
Step 4.1
To verify the inverse, check if and .
Step 4.2
Evaluate .
Tap for more steps...
Step 4.2.1
Set up the composite result function.
Step 4.2.2
Evaluate by substituting in the value of into .
Step 4.2.3
Move to the numerator using the negative exponent rule .
Step 4.2.4
Cancel the common factor of .
Tap for more steps...
Step 4.2.4.1
Cancel the common factor.
Step 4.2.4.2
Divide by .
Step 4.3
Evaluate .
Tap for more steps...
Step 4.3.1
Set up the composite result function.
Step 4.3.2
Evaluate by substituting in the value of into .
Step 4.3.3
Change the sign of the exponent by rewriting the base as its reciprocal.
Step 4.3.4
Cancel the common factor of .
Tap for more steps...
Step 4.3.4.1
Cancel the common factor.
Step 4.3.4.2
Rewrite the expression.
Step 4.4
Since and , then is the inverse of .