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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
To remove the radical on the left side of the equation, square both sides of the equation.
Step 3.3
Simplify each side of the equation.
Step 3.3.1
Use to rewrite as .
Step 3.3.2
Simplify the left side.
Step 3.3.2.1
Simplify .
Step 3.3.2.1.1
Multiply the exponents in .
Step 3.3.2.1.1.1
Apply the power rule and multiply exponents, .
Step 3.3.2.1.1.2
Cancel the common factor of .
Step 3.3.2.1.1.2.1
Cancel the common factor.
Step 3.3.2.1.1.2.2
Rewrite the expression.
Step 3.3.2.1.2
Simplify.
Step 3.4
Solve for .
Step 3.4.1
Add to both sides of the equation.
Step 3.4.2
Divide each term in by and simplify.
Step 3.4.2.1
Divide each term in by .
Step 3.4.2.2
Simplify the left side.
Step 3.4.2.2.1
Cancel the common factor of .
Step 3.4.2.2.1.1
Cancel the common factor.
Step 3.4.2.2.1.2
Divide by .
Step 3.4.2.3
Simplify the right side.
Step 3.4.2.3.1
Cancel the common factor of and .
Step 3.4.2.3.1.1
Factor out of .
Step 3.4.2.3.1.2
Cancel the common factors.
Step 3.4.2.3.1.2.1
Factor out of .
Step 3.4.2.3.1.2.2
Cancel the common factor.
Step 3.4.2.3.1.2.3
Rewrite the expression.
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
Simplify the numerator.
Step 5.2.3.1.1
Factor out of .
Step 5.2.3.1.1.1
Factor out of .
Step 5.2.3.1.1.2
Factor out of .
Step 5.2.3.1.1.3
Factor out of .
Step 5.2.3.1.2
Rewrite as .
Step 5.2.3.1.2.1
Use to rewrite as .
Step 5.2.3.1.2.2
Apply the power rule and multiply exponents, .
Step 5.2.3.1.2.3
Combine and .
Step 5.2.3.1.2.4
Cancel the common factor of .
Step 5.2.3.1.2.4.1
Cancel the common factor.
Step 5.2.3.1.2.4.2
Rewrite the expression.
Step 5.2.3.1.2.5
Simplify.
Step 5.2.3.1.3
Apply the distributive property.
Step 5.2.3.1.4
Multiply by .
Step 5.2.3.1.5
Multiply by .
Step 5.2.3.1.6
Factor out of .
Step 5.2.3.1.6.1
Factor out of .
Step 5.2.3.1.6.2
Factor out of .
Step 5.2.3.1.6.3
Factor out of .
Step 5.2.3.2
Cancel the common factors.
Step 5.2.3.2.1
Factor out of .
Step 5.2.3.2.2
Cancel the common factor.
Step 5.2.3.2.3
Rewrite the expression.
Step 5.2.4
Simplify terms.
Step 5.2.4.1
Combine the numerators over the common denominator.
Step 5.2.4.2
Combine the opposite terms in .
Step 5.2.4.2.1
Add and .
Step 5.2.4.2.2
Add and .
Step 5.2.4.3
Cancel the common factor of .
Step 5.2.4.3.1
Cancel the common factor.
Step 5.2.4.3.2
Divide by .
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
Factor out of .
Step 5.3.3.1
Factor out of .
Step 5.3.3.2
Factor out of .
Step 5.3.3.3
Factor out of .
Step 5.3.4
Apply the distributive property.
Step 5.3.5
Cancel the common factor of .
Step 5.3.5.1
Factor out of .
Step 5.3.5.2
Cancel the common factor.
Step 5.3.5.3
Rewrite the expression.
Step 5.3.6
Cancel the common factor of .
Step 5.3.6.1
Cancel the common factor.
Step 5.3.6.2
Rewrite the expression.
Step 5.3.7
Simplify by subtracting numbers.
Step 5.3.7.1
Subtract from .
Step 5.3.7.2
Add and .
Step 5.3.8
Combine and .
Step 5.3.9
Reduce the expression by cancelling the common factors.
Step 5.3.9.1
Reduce the expression by cancelling the common factors.
Step 5.3.9.1.1
Cancel the common factor.
Step 5.3.9.1.2
Rewrite the expression.
Step 5.3.9.2
Divide by .
Step 5.3.10
Pull terms out from under the radical, assuming positive real numbers.
Step 5.4
Since and , then is the inverse of .