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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
Subtract from both sides of the equation.
Step 3.3
Divide each term in by and simplify.
Step 3.3.1
Divide 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
Divide by .
Step 3.3.3
Simplify the right side.
Step 3.3.3.1
Divide by .
Step 3.4
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 3.5
Expand the left side.
Step 3.5.1
Expand by moving outside the logarithm.
Step 3.5.2
The natural logarithm of is .
Step 3.5.3
Multiply by .
Step 3.6
Add to both sides of the equation.
Step 4
Replace with to show the final answer.
Step 5
Step 5.1
To verify the inverse, check if and .
Step 5.2
Evaluate .
Step 5.2.1
Set up the composite result function.
Step 5.2.2
Evaluate by substituting in the value of into .
Step 5.2.3
Simplify each term.
Step 5.2.3.1
Cancel the common factor of and .
Step 5.2.3.1.1
Factor out of .
Step 5.2.3.1.2
Factor out of .
Step 5.2.3.1.3
Factor out of .
Step 5.2.3.1.4
Cancel the common factors.
Step 5.2.3.1.4.1
Factor out of .
Step 5.2.3.1.4.2
Cancel the common factor.
Step 5.2.3.1.4.3
Rewrite the expression.
Step 5.2.3.1.4.4
Divide by .
Step 5.2.3.2
Combine the opposite terms in .
Step 5.2.3.2.1
Subtract from .
Step 5.2.3.2.2
Add and .
Step 5.2.3.3
Use logarithm rules to move out of the exponent.
Step 5.2.3.4
The natural logarithm of is .
Step 5.2.3.5
Multiply by .
Step 5.2.4
Combine the opposite terms in .
Step 5.2.4.1
Add and .
Step 5.2.4.2
Add and .
Step 5.3
Evaluate .
Step 5.3.1
Set up the composite result function.
Step 5.3.2
Evaluate by substituting in the value of into .
Step 5.3.3
Combine the opposite terms in .
Step 5.3.3.1
Subtract from .
Step 5.3.3.2
Add and .
Step 5.3.4
Simplify each term.
Step 5.3.4.1
Exponentiation and log are inverse functions.
Step 5.3.4.2
Apply the distributive property.
Step 5.3.4.3
Cancel the common factor of .
Step 5.3.4.3.1
Cancel the common factor.
Step 5.3.4.3.2
Rewrite the expression.
Step 5.3.4.4
Multiply by .
Step 5.3.5
Combine the opposite terms in .
Step 5.3.5.1
Add and .
Step 5.3.5.2
Add and .
Step 5.4
Since and , then is the inverse of .