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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
Factor out of .
Step 2.3.1.1.1.1
Factor out of .
Step 2.3.1.1.1.2
Factor out of .
Step 2.3.1.1.1.3
Factor out of .
Step 2.3.1.1.2
Cancel the common factor of .
Step 2.3.1.1.2.1
Cancel the common factor.
Step 2.3.1.1.2.2
Rewrite the expression.
Step 2.3.2
Simplify the right side.
Step 2.3.2.1
Move to the left of .
Step 2.4
Solve for .
Step 2.4.1
To remove the radical on the left side of the equation, square both sides of the equation.
Step 2.4.2
Simplify each side of the equation.
Step 2.4.2.1
Use to rewrite as .
Step 2.4.2.2
Simplify the left side.
Step 2.4.2.2.1
Simplify .
Step 2.4.2.2.1.1
Multiply the exponents in .
Step 2.4.2.2.1.1.1
Apply the power rule and multiply exponents, .
Step 2.4.2.2.1.1.2
Cancel the common factor of .
Step 2.4.2.2.1.1.2.1
Cancel the common factor.
Step 2.4.2.2.1.1.2.2
Rewrite the expression.
Step 2.4.2.2.1.2
Apply the distributive property.
Step 2.4.2.2.1.3
Multiply.
Step 2.4.2.2.1.3.1
Multiply by .
Step 2.4.2.2.1.3.2
Simplify.
Step 2.4.2.3
Simplify the right side.
Step 2.4.2.3.1
Simplify .
Step 2.4.2.3.1.1
Apply the product rule to .
Step 2.4.2.3.1.2
Raise to the power of .
Step 2.4.3
Solve for .
Step 2.4.3.1
Subtract from both sides of the equation.
Step 2.4.3.2
Divide each term in by and simplify.
Step 2.4.3.2.1
Divide each term in by .
Step 2.4.3.2.2
Simplify the left side.
Step 2.4.3.2.2.1
Cancel the common factor of .
Step 2.4.3.2.2.1.1
Cancel the common factor.
Step 2.4.3.2.2.1.2
Divide by .
Step 2.4.3.2.3
Simplify the right side.
Step 2.4.3.2.3.1
Divide by .
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 each term.
Step 4.2.3.1
Factor out of .
Step 4.2.3.1.1
Factor out of .
Step 4.2.3.1.2
Factor out of .
Step 4.2.3.1.3
Factor out of .
Step 4.2.3.2
Simplify the numerator.
Step 4.2.3.2.1
Apply the product rule to .
Step 4.2.3.2.2
Simplify the numerator.
Step 4.2.3.2.2.1
Rewrite as .
Step 4.2.3.2.2.1.1
Use to rewrite as .
Step 4.2.3.2.2.1.2
Apply the power rule and multiply exponents, .
Step 4.2.3.2.2.1.3
Combine and .
Step 4.2.3.2.2.1.4
Cancel the common factor of .
Step 4.2.3.2.2.1.4.1
Cancel the common factor.
Step 4.2.3.2.2.1.4.2
Rewrite the expression.
Step 4.2.3.2.2.1.5
Simplify.
Step 4.2.3.2.2.2
Apply the distributive property.
Step 4.2.3.2.2.3
Multiply by .
Step 4.2.3.2.2.4
Factor out of .
Step 4.2.3.2.2.4.1
Factor out of .
Step 4.2.3.2.2.4.2
Factor out of .
Step 4.2.3.2.2.4.3
Factor out of .
Step 4.2.3.2.3
Raise to the power of .
Step 4.2.3.3
Combine and .
Step 4.2.3.4
Multiply by .
Step 4.2.3.5
Reduce the expression by cancelling the common factors.
Step 4.2.3.5.1
Reduce the expression by cancelling the common factors.
Step 4.2.3.5.1.1
Factor out of .
Step 4.2.3.5.1.2
Factor out of .
Step 4.2.3.5.1.3
Cancel the common factor.
Step 4.2.3.5.1.4
Rewrite the expression.
Step 4.2.3.5.2
Divide by .
Step 4.2.3.6
Cancel the common factor of .
Step 4.2.3.6.1
Cancel the common factor.
Step 4.2.3.6.2
Divide by .
Step 4.2.4
Combine the opposite terms in .
Step 4.2.4.1
Subtract from .
Step 4.2.4.2
Add 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
Simplify the numerator.
Step 4.3.3.1
Factor out of .
Step 4.3.3.1.1
Factor out of .
Step 4.3.3.1.2
Factor out of .
Step 4.3.3.2
To write as a fraction with a common denominator, multiply by .
Step 4.3.3.3
Combine and .
Step 4.3.3.4
Combine the numerators over the common denominator.
Step 4.3.3.5
To write as a fraction with a common denominator, multiply by .
Step 4.3.3.6
Combine and .
Step 4.3.3.7
Combine the numerators over the common denominator.
Step 4.3.3.8
Reorder terms.
Step 4.3.3.9
Rewrite in a factored form.
Step 4.3.3.9.1
Multiply by .
Step 4.3.3.9.2
Multiply by .
Step 4.3.3.9.3
Subtract from .
Step 4.3.3.9.4
Add and .
Step 4.3.3.10
Combine and .
Step 4.3.3.11
Multiply by .
Step 4.3.3.12
Reduce the expression by cancelling the common factors.
Step 4.3.3.12.1
Reduce the expression by cancelling the common factors.
Step 4.3.3.12.1.1
Factor out of .
Step 4.3.3.12.1.2
Factor out of .
Step 4.3.3.12.1.3
Cancel the common factor.
Step 4.3.3.12.1.4
Rewrite the expression.
Step 4.3.3.12.2
Divide by .
Step 4.3.3.13
Rewrite as .
Step 4.3.3.14
Pull terms out from under the radical, assuming positive real numbers.
Step 4.3.4
Cancel the common factor of .
Step 4.3.4.1
Cancel the common factor.
Step 4.3.4.2
Divide by .
Step 4.4
Since and , then is the inverse of .