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Trigonometry Examples
Step 1
Interchange the variables.
Step 2
Step 2.1
Rewrite the equation as .
Step 2.2
Multiply both sides by .
Step 2.3
Simplify.
Step 2.3.1
Simplify the left side.
Step 2.3.1.1
Simplify .
Step 2.3.1.1.1
Cancel the common factor of .
Step 2.3.1.1.1.1
Cancel the common factor.
Step 2.3.1.1.1.2
Rewrite the expression.
Step 2.3.1.1.2
Simplify each term.
Step 2.3.1.1.2.1
Since is an odd function, rewrite as .
Step 2.3.1.1.2.2
Multiply .
Step 2.3.1.1.2.2.1
Multiply by .
Step 2.3.1.1.2.2.2
Multiply by .
Step 2.3.2
Simplify the right side.
Step 2.3.2.1
Simplify .
Step 2.3.2.1.1
Apply the distributive property.
Step 2.3.2.1.2
Multiply by .
Step 2.4
Solve for .
Step 2.4.1
Substitute for .
Step 2.4.2
Subtract from both sides of the equation.
Step 2.4.3
Subtract from both sides of the equation.
Step 2.4.4
Factor out of .
Step 2.4.4.1
Factor out of .
Step 2.4.4.2
Factor out of .
Step 2.4.4.3
Factor out of .
Step 2.4.5
Divide each term in by and simplify.
Step 2.4.5.1
Divide each term in by .
Step 2.4.5.2
Simplify the left side.
Step 2.4.5.2.1
Cancel the common factor of .
Step 2.4.5.2.1.1
Cancel the common factor.
Step 2.4.5.2.1.2
Divide by .
Step 2.4.5.3
Simplify the right side.
Step 2.4.5.3.1
Combine the numerators over the common denominator.
Step 2.4.5.3.2
Cancel the common factor of and .
Step 2.4.5.3.2.1
Factor out of .
Step 2.4.5.3.2.2
Rewrite as .
Step 2.4.5.3.2.3
Factor out of .
Step 2.4.5.3.2.4
Reorder terms.
Step 2.4.5.3.2.5
Cancel the common factor.
Step 2.4.5.3.2.6
Divide by .
Step 2.4.6
Substitute for .
Step 2.4.7
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 2.4.8
Simplify the right side.
Step 2.4.8.1
The exact value of is .
Step 2.4.9
The cotangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 2.4.10
Simplify the expression to find the second solution.
Step 2.4.10.1
Add to .
Step 2.4.10.2
The resulting angle of is positive and coterminal with .
Step 2.4.11
Find the period of .
Step 2.4.11.1
The period of the function can be calculated using .
Step 2.4.11.2
Replace with in the formula for period.
Step 2.4.11.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.4.11.4
Divide by .
Step 2.4.12
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 2.5
Consolidate the answers.
, for any integer
, for any integer
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
Reorder and .
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
Apply the distributive property.
Step 4.4
Since and , then is the inverse of .