Trigonometry Examples

Find the Inverse y=sin( square root of x)+2
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Subtract from both sides of the equation.
Step 2.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2.4
To remove the radical on the left side of the equation, square both sides of the equation.
Step 2.5
Simplify each side of the equation.
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Step 2.5.1
Use to rewrite as .
Step 2.5.2
Simplify the left side.
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Step 2.5.2.1
Simplify .
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Step 2.5.2.1.1
Multiply the exponents in .
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Step 2.5.2.1.1.1
Apply the power rule and multiply exponents, .
Step 2.5.2.1.1.2
Cancel the common factor of .
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Step 2.5.2.1.1.2.1
Cancel the common factor.
Step 2.5.2.1.1.2.2
Rewrite the expression.
Step 2.5.2.1.2
Simplify.
Step 3
Replace with to show the final answer.
Step 4
Verify if is the inverse of .
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Step 4.1
To verify the inverse, check if and .
Step 4.2
Evaluate .
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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
Combine the opposite terms in .
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Step 4.2.3.1
Subtract from .
Step 4.2.3.2
Add and .
Step 4.3
Evaluate .
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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
Remove parentheses.
Step 4.3.4
Simplify each term.
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Step 4.3.4.1
Pull terms out from under the radical, assuming positive real numbers.
Step 4.3.4.2
The functions sine and arcsine are inverses.
Step 4.3.5
Combine the opposite terms in .
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Step 4.3.5.1
Add and .
Step 4.3.5.2
Add and .
Step 4.4
Since and , then is the inverse of .