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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
Add to both sides of the equation.
Step 3.3
To remove the radical on the left side of the equation, cube both sides of the equation.
Step 3.4
Simplify each side of the equation.
Step 3.4.1
Use to rewrite as .
Step 3.4.2
Simplify the left side.
Step 3.4.2.1
Simplify .
Step 3.4.2.1.1
Multiply the exponents in .
Step 3.4.2.1.1.1
Apply the power rule and multiply exponents, .
Step 3.4.2.1.1.2
Cancel the common factor of .
Step 3.4.2.1.1.2.1
Cancel the common factor.
Step 3.4.2.1.1.2.2
Rewrite the expression.
Step 3.4.2.1.2
Simplify.
Step 3.4.3
Simplify the right side.
Step 3.4.3.1
Simplify .
Step 3.4.3.1.1
Use the Binomial Theorem.
Step 3.4.3.1.2
Simplify each term.
Step 3.4.3.1.2.1
Multiply by .
Step 3.4.3.1.2.2
Raise to the power of .
Step 3.4.3.1.2.3
Multiply by .
Step 3.4.3.1.2.4
Raise to the power of .
Step 3.5
Solve for .
Step 3.5.1
Multiply both sides of the equation by .
Step 3.5.2
Simplify both sides of the equation.
Step 3.5.2.1
Simplify the left side.
Step 3.5.2.1.1
Cancel the common factor of .
Step 3.5.2.1.1.1
Cancel the common factor.
Step 3.5.2.1.1.2
Rewrite the expression.
Step 3.5.2.2
Simplify the right side.
Step 3.5.2.2.1
Simplify .
Step 3.5.2.2.1.1
Apply the distributive property.
Step 3.5.2.2.1.2
Simplify.
Step 3.5.2.2.1.2.1
Multiply by .
Step 3.5.2.2.1.2.2
Multiply by .
Step 3.5.2.2.1.2.3
Multiply by .
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 each term.
Step 5.2.3.1.1
Rewrite as .
Step 5.2.3.1.2
Multiply by .
Step 5.2.3.1.3
Combine and simplify the denominator.
Step 5.2.3.1.3.1
Multiply by .
Step 5.2.3.1.3.2
Raise to the power of .
Step 5.2.3.1.3.3
Use the power rule to combine exponents.
Step 5.2.3.1.3.4
Add and .
Step 5.2.3.1.3.5
Rewrite as .
Step 5.2.3.1.3.5.1
Use to rewrite as .
Step 5.2.3.1.3.5.2
Apply the power rule and multiply exponents, .
Step 5.2.3.1.3.5.3
Combine and .
Step 5.2.3.1.3.5.4
Cancel the common factor of .
Step 5.2.3.1.3.5.4.1
Cancel the common factor.
Step 5.2.3.1.3.5.4.2
Rewrite the expression.
Step 5.2.3.1.3.5.5
Evaluate the exponent.
Step 5.2.3.1.4
Simplify the numerator.
Step 5.2.3.1.4.1
Rewrite as .
Step 5.2.3.1.4.2
Raise to the power of .
Step 5.2.3.1.5
Combine using the product rule for radicals.
Step 5.2.3.2
Use the Binomial Theorem.
Step 5.2.3.3
Simplify each term.
Step 5.2.3.3.1
Apply the product rule to .
Step 5.2.3.3.2
Simplify the numerator.
Step 5.2.3.3.2.1
Rewrite as .
Step 5.2.3.3.2.1.1
Use to rewrite as .
Step 5.2.3.3.2.1.2
Apply the power rule and multiply exponents, .
Step 5.2.3.3.2.1.3
Combine and .
Step 5.2.3.3.2.1.4
Cancel the common factor of .
Step 5.2.3.3.2.1.4.1
Cancel the common factor.
Step 5.2.3.3.2.1.4.2
Rewrite the expression.
Step 5.2.3.3.2.1.5
Simplify.
Step 5.2.3.3.2.2
Move to the left of .
Step 5.2.3.3.3
Raise to the power of .
Step 5.2.3.3.4
Cancel the common factor of and .
Step 5.2.3.3.4.1
Factor out of .
Step 5.2.3.3.4.2
Cancel the common factors.
Step 5.2.3.3.4.2.1
Factor out of .
Step 5.2.3.3.4.2.2
Cancel the common factor.
Step 5.2.3.3.4.2.3
Rewrite the expression.
Step 5.2.3.3.5
Apply the product rule to .
Step 5.2.3.3.6
Simplify the numerator.
Step 5.2.3.3.6.1
Rewrite as .
Step 5.2.3.3.6.2
Apply the product rule to .
Step 5.2.3.3.6.3
Raise to the power of .
Step 5.2.3.3.6.4
Rewrite as .
Step 5.2.3.3.6.4.1
Factor out of .
Step 5.2.3.3.6.4.2
Rewrite as .
Step 5.2.3.3.6.4.3
Reorder and .
Step 5.2.3.3.6.4.4
Add parentheses.
Step 5.2.3.3.6.5
Pull terms out from under the radical.
Step 5.2.3.3.7
Raise to the power of .
Step 5.2.3.3.8
Cancel the common factor of and .
Step 5.2.3.3.8.1
Factor out of .
Step 5.2.3.3.8.2
Cancel the common factors.
Step 5.2.3.3.8.2.1
Factor out of .
Step 5.2.3.3.8.2.2
Cancel the common factor.
Step 5.2.3.3.8.2.3
Rewrite the expression.
Step 5.2.3.3.9
Combine and .
Step 5.2.3.3.10
Move to the left of .
Step 5.2.3.3.11
Multiply .
Step 5.2.3.3.11.1
Combine and .
Step 5.2.3.3.11.2
Multiply by .
Step 5.2.3.3.12
Move the negative in front of the fraction.
Step 5.2.3.3.13
Combine and .
Step 5.2.3.3.14
Move to the left of .
Step 5.2.3.3.15
Raise to the power of .
Step 5.2.3.3.16
Multiply .
Step 5.2.3.3.16.1
Combine and .
Step 5.2.3.3.16.2
Multiply by .
Step 5.2.3.3.17
Raise to the power of .
Step 5.2.3.4
Apply the distributive property.
Step 5.2.3.5
Simplify.
Step 5.2.3.5.1
Cancel the common factor of .
Step 5.2.3.5.1.1
Cancel the common factor.
Step 5.2.3.5.1.2
Rewrite the expression.
Step 5.2.3.5.2
Cancel the common factor of .
Step 5.2.3.5.2.1
Move the leading negative in into the numerator.
Step 5.2.3.5.2.2
Cancel the common factor.
Step 5.2.3.5.2.3
Rewrite the expression.
Step 5.2.3.5.3
Cancel the common factor of .
Step 5.2.3.5.3.1
Cancel the common factor.
Step 5.2.3.5.3.2
Rewrite the expression.
Step 5.2.3.5.4
Multiply by .
Step 5.2.3.6
Simplify each term.
Step 5.2.3.6.1
Rewrite as .
Step 5.2.3.6.2
Multiply by .
Step 5.2.3.6.3
Combine and simplify the denominator.
Step 5.2.3.6.3.1
Multiply by .
Step 5.2.3.6.3.2
Raise to the power of .
Step 5.2.3.6.3.3
Use the power rule to combine exponents.
Step 5.2.3.6.3.4
Add and .
Step 5.2.3.6.3.5
Rewrite as .
Step 5.2.3.6.3.5.1
Use to rewrite as .
Step 5.2.3.6.3.5.2
Apply the power rule and multiply exponents, .
Step 5.2.3.6.3.5.3
Combine and .
Step 5.2.3.6.3.5.4
Cancel the common factor of .
Step 5.2.3.6.3.5.4.1
Cancel the common factor.
Step 5.2.3.6.3.5.4.2
Rewrite the expression.
Step 5.2.3.6.3.5.5
Evaluate the exponent.
Step 5.2.3.6.4
Simplify the numerator.
Step 5.2.3.6.4.1
Rewrite as .
Step 5.2.3.6.4.2
Raise to the power of .
Step 5.2.3.6.5
Combine using the product rule for radicals.
Step 5.2.3.7
Rewrite as .
Step 5.2.3.8
Expand using the FOIL Method.
Step 5.2.3.8.1
Apply the distributive property.
Step 5.2.3.8.2
Apply the distributive property.
Step 5.2.3.8.3
Apply the distributive property.
Step 5.2.3.9
Simplify and combine like terms.
Step 5.2.3.9.1
Simplify each term.
Step 5.2.3.9.1.1
Multiply .
Step 5.2.3.9.1.1.1
Multiply by .
Step 5.2.3.9.1.1.2
Raise to the power of .
Step 5.2.3.9.1.1.3
Raise to the power of .
Step 5.2.3.9.1.1.4
Use the power rule to combine exponents.
Step 5.2.3.9.1.1.5
Add and .
Step 5.2.3.9.1.1.6
Multiply by .
Step 5.2.3.9.1.2
Simplify the numerator.
Step 5.2.3.9.1.2.1
Rewrite as .
Step 5.2.3.9.1.2.2
Apply the product rule to .
Step 5.2.3.9.1.2.3
Raise to the power of .
Step 5.2.3.9.1.2.4
Rewrite as .
Step 5.2.3.9.1.2.4.1
Factor out of .
Step 5.2.3.9.1.2.4.2
Rewrite as .
Step 5.2.3.9.1.2.4.3
Reorder and .
Step 5.2.3.9.1.2.4.4
Add parentheses.
Step 5.2.3.9.1.2.5
Pull terms out from under the radical.
Step 5.2.3.9.1.3
Cancel the common factor of and .
Step 5.2.3.9.1.3.1
Factor out of .
Step 5.2.3.9.1.3.2
Cancel the common factors.
Step 5.2.3.9.1.3.2.1
Factor out of .
Step 5.2.3.9.1.3.2.2
Cancel the common factor.
Step 5.2.3.9.1.3.2.3
Rewrite the expression.
Step 5.2.3.9.1.4
Combine and .
Step 5.2.3.9.1.5
Move to the left of .
Step 5.2.3.9.1.6
Move to the left of .
Step 5.2.3.9.1.7
Move the negative in front of the fraction.
Step 5.2.3.9.1.8
Combine and .
Step 5.2.3.9.1.9
Move to the left of .
Step 5.2.3.9.1.10
Move the negative in front of the fraction.
Step 5.2.3.9.1.11
Multiply by .
Step 5.2.3.9.2
Subtract from .
Step 5.2.3.10
Simplify each term.
Step 5.2.3.10.1
Multiply .
Step 5.2.3.10.1.1
Combine and .
Step 5.2.3.10.1.2
Multiply by .
Step 5.2.3.10.2
Move the negative in front of the fraction.
Step 5.2.3.11
Apply the distributive property.
Step 5.2.3.12
Simplify.
Step 5.2.3.12.1
Cancel the common factor of .
Step 5.2.3.12.1.1
Factor out of .
Step 5.2.3.12.1.2
Cancel the common factor.
Step 5.2.3.12.1.3
Rewrite the expression.
Step 5.2.3.12.2
Cancel the common factor of .
Step 5.2.3.12.2.1
Move the leading negative in into the numerator.
Step 5.2.3.12.2.2
Factor out of .
Step 5.2.3.12.2.3
Cancel the common factor.
Step 5.2.3.12.2.4
Rewrite the expression.
Step 5.2.3.12.3
Multiply by .
Step 5.2.3.12.4
Multiply by .
Step 5.2.3.13
Simplify each term.
Step 5.2.3.13.1
Rewrite as .
Step 5.2.3.13.2
Multiply by .
Step 5.2.3.13.3
Combine and simplify the denominator.
Step 5.2.3.13.3.1
Multiply by .
Step 5.2.3.13.3.2
Raise to the power of .
Step 5.2.3.13.3.3
Use the power rule to combine exponents.
Step 5.2.3.13.3.4
Add and .
Step 5.2.3.13.3.5
Rewrite as .
Step 5.2.3.13.3.5.1
Use to rewrite as .
Step 5.2.3.13.3.5.2
Apply the power rule and multiply exponents, .
Step 5.2.3.13.3.5.3
Combine and .
Step 5.2.3.13.3.5.4
Cancel the common factor of .
Step 5.2.3.13.3.5.4.1
Cancel the common factor.
Step 5.2.3.13.3.5.4.2
Rewrite the expression.
Step 5.2.3.13.3.5.5
Evaluate the exponent.
Step 5.2.3.13.4
Simplify the numerator.
Step 5.2.3.13.4.1
Rewrite as .
Step 5.2.3.13.4.2
Raise to the power of .
Step 5.2.3.13.5
Combine using the product rule for radicals.
Step 5.2.3.14
Apply the distributive property.
Step 5.2.3.15
Cancel the common factor of .
Step 5.2.3.15.1
Factor out of .
Step 5.2.3.15.2
Cancel the common factor.
Step 5.2.3.15.3
Rewrite the expression.
Step 5.2.3.16
Multiply by .
Step 5.2.4
Combine the opposite terms in .
Step 5.2.4.1
Subtract from .
Step 5.2.4.2
Add and .
Step 5.2.4.3
Add and .
Step 5.2.4.4
Add and .
Step 5.2.5
Add and .
Step 5.2.5.1
Reorder and .
Step 5.2.5.2
Add and .
Step 5.2.6
Add and .
Step 5.2.7
Subtract from .
Step 5.2.8
Add and .
Step 5.2.8.1
Reorder and .
Step 5.2.8.2
Add and .
Step 5.2.9
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
Simplify each term.
Step 5.3.3.1
Factor out of .
Step 5.3.3.1.1
Factor out of .
Step 5.3.3.1.2
Factor out of .
Step 5.3.3.1.3
Factor out of .
Step 5.3.3.1.4
Factor out of .
Step 5.3.3.1.5
Factor out of .
Step 5.3.3.1.6
Factor out of .
Step 5.3.3.1.7
Factor out of .
Step 5.3.3.2
Reduce the expression by cancelling the common factors.
Step 5.3.3.2.1
Reduce the expression by cancelling the common factors.
Step 5.3.3.2.1.1
Cancel the common factor.
Step 5.3.3.2.1.2
Rewrite the expression.
Step 5.3.3.2.2
Divide by .
Step 5.3.3.3
Make each term match the terms from the binomial theorem formula.
Step 5.3.3.4
Factor using the binomial theorem.
Step 5.3.3.5
Pull terms out from under the radical, assuming real numbers.
Step 5.3.4
Combine the opposite terms in .
Step 5.3.4.1
Subtract from .
Step 5.3.4.2
Add and .
Step 5.4
Since and , then is the inverse of .