Trigonometry Examples

Find the Inverse y=1/3*sin(1/2x)
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 the left side.
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Step 2.3.1
Simplify .
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Step 2.3.1.1
Combine and .
Step 2.3.1.2
Cancel the common factor of .
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Step 2.3.1.2.1
Cancel the common factor.
Step 2.3.1.2.2
Rewrite the expression.
Step 2.4
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2.5
Simplify the left side.
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Step 2.5.1
Combine and .
Step 2.6
Multiply both sides of the equation by .
Step 2.7
Simplify the left side.
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Step 2.7.1
Cancel the common factor of .
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Step 2.7.1.1
Cancel the common factor.
Step 2.7.1.2
Rewrite the expression.
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 and .
Step 4.2.4
Combine and .
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
Rewrite the expression.
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
Cancel the common factor of .
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Step 4.3.3.1
Factor out of .
Step 4.3.3.2
Cancel the common factor.
Step 4.3.3.3
Rewrite the expression.
Step 4.3.4
The functions sine and arcsine are inverses.
Step 4.3.5
Cancel the common factor of .
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Step 4.3.5.1
Factor out of .
Step 4.3.5.2
Cancel the common factor.
Step 4.3.5.3
Rewrite the expression.
Step 4.4
Since and , then is the inverse of .