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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
Move the negative in front of the fraction.
Step 3.4
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 3.5
Solve for .
Step 3.5.1
Rewrite the equation as .
Step 3.5.2
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
Combine the numerators over the common denominator.
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
Simplify by moving inside the logarithm.
Step 5.2.3.4
Expand by moving outside the logarithm.
Step 5.2.3.5
Cancel the common factor of .
Step 5.2.3.5.1
Cancel the common factor.
Step 5.2.3.5.2
Divide by .
Step 5.2.3.6
Exponentiation and log are inverse functions.
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
Use logarithm rules to move out of the exponent.
Step 5.3.4.2
Logarithm base of is .
Step 5.3.4.3
Multiply by .
Step 5.3.4.4
Apply the distributive property.
Step 5.3.4.5
Cancel the common factor of .
Step 5.3.4.5.1
Cancel the common factor.
Step 5.3.4.5.2
Rewrite the expression.
Step 5.3.4.6
Cancel the common factor of .
Step 5.3.4.6.1
Move the leading negative in into the numerator.
Step 5.3.4.6.2
Cancel the common factor.
Step 5.3.4.6.3
Rewrite the expression.
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 .