Trigonometry Examples

Find the Inverse y=arcsin((x+3)/4)
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Take the inverse arcsine of both sides of the equation to extract from inside the arcsine.
Step 2.3
Simplify the left side.
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Step 2.3.1
Split the fraction into two fractions.
Step 2.4
Subtract from both sides of the equation.
Step 2.5
Multiply both sides of the equation by .
Step 2.6
Simplify both sides of the equation.
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Step 2.6.1
Simplify the left side.
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Step 2.6.1.1
Cancel the common factor of .
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Step 2.6.1.1.1
Cancel the common factor.
Step 2.6.1.1.2
Rewrite the expression.
Step 2.6.2
Simplify the right side.
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Step 2.6.2.1
Simplify .
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Step 2.6.2.1.1
Apply the distributive property.
Step 2.6.2.1.2
Cancel the common factor of .
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Step 2.6.2.1.2.1
Move the leading negative in into the numerator.
Step 2.6.2.1.2.2
Cancel the common factor.
Step 2.6.2.1.2.3
Rewrite the expression.
Step 2.7
Subtract from both sides of the equation.
Step 2.8
Multiply both sides of the equation by .
Step 2.9
Simplify both sides of the equation.
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Step 2.9.1
Simplify the left side.
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Step 2.9.1.1
Cancel the common factor of .
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Step 2.9.1.1.1
Cancel the common factor.
Step 2.9.1.1.2
Rewrite the expression.
Step 2.9.2
Simplify the right side.
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Step 2.9.2.1
Simplify .
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Step 2.9.2.1.1
Apply the distributive property.
Step 2.9.2.1.2
Cancel the common factor of .
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Step 2.9.2.1.2.1
Move the leading negative in into the numerator.
Step 2.9.2.1.2.2
Cancel the common factor.
Step 2.9.2.1.2.3
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
Simplify each term.
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Step 4.2.3.1
The functions sine and arcsine are inverses.
Step 4.2.3.2
Cancel the common factor of .
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Step 4.2.3.2.1
Cancel the common factor.
Step 4.2.3.2.2
Rewrite the expression.
Step 4.2.4
Combine the opposite terms in .
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Step 4.2.4.1
Subtract from .
Step 4.2.4.2
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
Simplify the numerator.
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Step 4.3.3.1
Add and .
Step 4.3.3.2
Add and .
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 .