Trigonometry Examples

Find the Inverse (sin(x) square root of 2)/2
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Multiply both sides of the equation by .
Step 2.3
Simplify both sides of the equation.
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Step 2.3.1
Simplify the left side.
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Step 2.3.1.1
Simplify .
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Step 2.3.1.1.1
Cancel the common factor of .
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Step 2.3.1.1.1.1
Cancel the common factor.
Step 2.3.1.1.1.2
Rewrite the expression.
Step 2.3.1.1.2
Cancel the common factor of .
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Step 2.3.1.1.2.1
Factor out of .
Step 2.3.1.1.2.2
Cancel the common factor.
Step 2.3.1.1.2.3
Rewrite the expression.
Step 2.3.2
Simplify the right side.
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Step 2.3.2.1
Simplify .
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Step 2.3.2.1.1
Multiply by .
Step 2.3.2.1.2
Combine and simplify the denominator.
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Step 2.3.2.1.2.1
Multiply by .
Step 2.3.2.1.2.2
Raise to the power of .
Step 2.3.2.1.2.3
Raise to the power of .
Step 2.3.2.1.2.4
Use the power rule to combine exponents.
Step 2.3.2.1.2.5
Add and .
Step 2.3.2.1.2.6
Rewrite as .
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Step 2.3.2.1.2.6.1
Use to rewrite as .
Step 2.3.2.1.2.6.2
Apply the power rule and multiply exponents, .
Step 2.3.2.1.2.6.3
Combine and .
Step 2.3.2.1.2.6.4
Cancel the common factor of .
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Step 2.3.2.1.2.6.4.1
Cancel the common factor.
Step 2.3.2.1.2.6.4.2
Rewrite the expression.
Step 2.3.2.1.2.6.5
Evaluate the exponent.
Step 2.3.2.1.3
Cancel the common factor of .
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Step 2.3.2.1.3.1
Cancel the common factor.
Step 2.3.2.1.3.2
Divide by .
Step 2.4
Take the inverse sine of both sides of the equation to extract from inside the sine.
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
Multiply .
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Step 4.2.3.1
Combine and .
Step 4.2.3.2
Raise to the power of .
Step 4.2.3.3
Raise to the power of .
Step 4.2.3.4
Use the power rule to combine exponents.
Step 4.2.3.5
Add and .
Step 4.2.4
Rewrite as .
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Step 4.2.4.1
Use to rewrite as .
Step 4.2.4.2
Apply the power rule and multiply exponents, .
Step 4.2.4.3
Combine and .
Step 4.2.4.4
Cancel the common factor of .
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Step 4.2.4.4.1
Cancel the common factor.
Step 4.2.4.4.2
Rewrite the expression.
Step 4.2.4.5
Evaluate the exponent.
Step 4.2.5
Cancel the common factor of .
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Step 4.2.5.1
Cancel the common factor.
Step 4.2.5.2
Divide by .
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
Simplify the numerator.
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Step 4.3.3.1
The functions sine and arcsine are inverses.
Step 4.3.3.2
Combine exponents.
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Step 4.3.3.2.1
Raise to the power of .
Step 4.3.3.2.2
Raise to the power of .
Step 4.3.3.2.3
Use the power rule to combine exponents.
Step 4.3.3.2.4
Add and .
Step 4.3.3.3
Rewrite as .
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Step 4.3.3.3.1
Use to rewrite as .
Step 4.3.3.3.2
Apply the power rule and multiply exponents, .
Step 4.3.3.3.3
Combine and .
Step 4.3.3.3.4
Cancel the common factor of .
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Step 4.3.3.3.4.1
Cancel the common factor.
Step 4.3.3.3.4.2
Rewrite the expression.
Step 4.3.3.3.5
Evaluate the exponent.
Step 4.3.4
Cancel the common factor of .
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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 .