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Trigonometry Examples
Step 1
Interchange the variables.
Step 2
Step 2.1
Rewrite the equation as .
Step 2.2
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 2.3
Simplify the left side.
Step 2.3.1
Simplify .
Step 2.3.1.1
Combine and .
Step 2.3.1.2
Move to the left of .
Step 2.4
Multiply both sides of the equation by .
Step 2.5
Simplify both sides of the equation.
Step 2.5.1
Simplify the left side.
Step 2.5.1.1
Simplify .
Step 2.5.1.1.1
Cancel the common factor of .
Step 2.5.1.1.1.1
Move the leading negative in into the numerator.
Step 2.5.1.1.1.2
Move the leading negative in into the numerator.
Step 2.5.1.1.1.3
Factor out of .
Step 2.5.1.1.1.4
Cancel the common factor.
Step 2.5.1.1.1.5
Rewrite the expression.
Step 2.5.1.1.2
Cancel the common factor of .
Step 2.5.1.1.2.1
Factor out of .
Step 2.5.1.1.2.2
Cancel the common factor.
Step 2.5.1.1.2.3
Rewrite the expression.
Step 2.5.1.1.3
Multiply.
Step 2.5.1.1.3.1
Multiply by .
Step 2.5.1.1.3.2
Multiply by .
Step 2.5.2
Simplify the right side.
Step 2.5.2.1
Simplify .
Step 2.5.2.1.1
Combine and .
Step 2.5.2.1.2
Move to the left of .
Step 3
Replace with to show the final answer.
Step 4
Step 4.1
To verify the inverse, check if and .
Step 4.2
Evaluate .
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
Simplify the numerator.
Step 4.2.3.1
Combine and .
Step 4.2.3.2
Move to the left of .
Step 4.2.3.3
Since is an odd function, rewrite as .
Step 4.3
Evaluate .
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
Cancel the common factor of .
Step 4.3.3.1
Move the leading negative in into the numerator.
Step 4.3.3.2
Move the leading negative in into the numerator.
Step 4.3.3.3
Factor out of .
Step 4.3.3.4
Cancel the common factor.
Step 4.3.3.5
Rewrite the expression.
Step 4.3.4
Cancel the common factor of .
Step 4.3.4.1
Factor out of .
Step 4.3.4.2
Cancel the common factor.
Step 4.3.4.3
Rewrite the expression.
Step 4.3.5
Multiply.
Step 4.3.5.1
Multiply by .
Step 4.3.5.2
Multiply by .
Step 4.3.6
The functions tangent and arctangent are inverses.
Step 4.4
Since and , then is the inverse of .