Trigonometry Examples

Find the Inverse f(x)=2sin(4x+1)^3
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Solve for .
Tap for more steps...
Step 3.1
Rewrite the equation as .
Step 3.2
Divide each term in by and simplify.
Tap for more steps...
Step 3.2.1
Divide each term in by .
Step 3.2.2
Simplify the left side.
Tap for more steps...
Step 3.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 3.2.2.1.1
Cancel the common factor.
Step 3.2.2.1.2
Divide by .
Step 3.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.4
Simplify .
Tap for more steps...
Step 3.4.1
Rewrite as .
Step 3.4.2
Multiply by .
Step 3.4.3
Combine and simplify the denominator.
Tap for more steps...
Step 3.4.3.1
Multiply by .
Step 3.4.3.2
Raise to the power of .
Step 3.4.3.3
Use the power rule to combine exponents.
Step 3.4.3.4
Add and .
Step 3.4.3.5
Rewrite as .
Tap for more steps...
Step 3.4.3.5.1
Use to rewrite as .
Step 3.4.3.5.2
Apply the power rule and multiply exponents, .
Step 3.4.3.5.3
Combine and .
Step 3.4.3.5.4
Cancel the common factor of .
Tap for more steps...
Step 3.4.3.5.4.1
Cancel the common factor.
Step 3.4.3.5.4.2
Rewrite the expression.
Step 3.4.3.5.5
Evaluate the exponent.
Step 3.4.4
Simplify the numerator.
Tap for more steps...
Step 3.4.4.1
Rewrite as .
Step 3.4.4.2
Raise to the power of .
Step 3.4.5
Simplify with factoring out.
Tap for more steps...
Step 3.4.5.1
Combine using the product rule for radicals.
Step 3.4.5.2
Reorder factors in .
Step 3.5
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 3.6
Subtract from both sides of the equation.
Step 3.7
Divide each term in by and simplify.
Tap for more steps...
Step 3.7.1
Divide each term in by .
Step 3.7.2
Simplify the left side.
Tap for more steps...
Step 3.7.2.1
Cancel the common factor of .
Tap for more steps...
Step 3.7.2.1.1
Cancel the common factor.
Step 3.7.2.1.2
Divide by .
Step 3.7.3
Simplify the right side.
Tap for more steps...
Step 3.7.3.1
Move the negative in front of the fraction.
Step 4
Replace with to show the final answer.
Step 5
Verify if is the inverse of .
Tap for more steps...
Step 5.1
To verify the inverse, check if and .
Step 5.2
Evaluate .
Tap for more steps...
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.
Tap for more steps...
Step 5.2.3.1
Simplify the numerator.
Tap for more steps...
Step 5.2.3.1.1
Multiply by .
Step 5.2.3.1.2
Rewrite as .
Step 5.2.3.1.3
Pull terms out from under the radical, assuming real numbers.
Step 5.2.3.2
Cancel the common factor of .
Tap for more steps...
Step 5.2.3.2.1
Cancel the common factor.
Step 5.2.3.2.2
Divide by .
Step 5.3
Evaluate .
Tap for more steps...
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.
Tap for more steps...
Step 5.3.3.1
Apply the distributive property.
Step 5.3.3.2
Cancel the common factor of .
Tap for more steps...
Step 5.3.3.2.1
Cancel the common factor.
Step 5.3.3.2.2
Rewrite the expression.
Step 5.3.3.3
Cancel the common factor of .
Tap for more steps...
Step 5.3.3.3.1
Move the leading negative in into the numerator.
Step 5.3.3.3.2
Cancel the common factor.
Step 5.3.3.3.3
Rewrite the expression.
Step 5.3.4
Combine the opposite terms in .
Tap for more steps...
Step 5.3.4.1
Add and .
Step 5.3.4.2
Add and .
Step 5.3.5
The functions sine and arcsine are inverses.
Step 5.3.6
Apply the product rule to .
Step 5.3.7
Rewrite as .
Tap for more steps...
Step 5.3.7.1
Use to rewrite as .
Step 5.3.7.2
Apply the power rule and multiply exponents, .
Step 5.3.7.3
Combine and .
Step 5.3.7.4
Cancel the common factor of .
Tap for more steps...
Step 5.3.7.4.1
Cancel the common factor.
Step 5.3.7.4.2
Rewrite the expression.
Step 5.3.7.5
Simplify.
Step 5.3.8
Raise to the power of .
Step 5.3.9
Cancel the common factor of .
Tap for more steps...
Step 5.3.9.1
Factor out of .
Step 5.3.9.2
Cancel the common factor.
Step 5.3.9.3
Rewrite the expression.
Step 5.3.10
Cancel the common factor of .
Tap for more steps...
Step 5.3.10.1
Cancel the common factor.
Step 5.3.10.2
Divide by .
Step 5.4
Since and , then is the inverse of .