Trigonometry Examples

Find the Inverse -4csc(1/2x)+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
Divide each term in by and simplify.
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Step 2.3.1
Divide each term in by .
Step 2.3.2
Simplify the left side.
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Step 2.3.2.1
Cancel the common factor of .
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Step 2.3.2.1.1
Cancel the common factor.
Step 2.3.2.1.2
Divide by .
Step 2.3.3
Simplify the right side.
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Step 2.3.3.1
Simplify each term.
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Step 2.3.3.1.1
Move the negative in front of the fraction.
Step 2.3.3.1.2
Cancel the common factor of and .
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Step 2.3.3.1.2.1
Factor out of .
Step 2.3.3.1.2.2
Cancel the common factors.
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Step 2.3.3.1.2.2.1
Factor out of .
Step 2.3.3.1.2.2.2
Cancel the common factor.
Step 2.3.3.1.2.2.3
Rewrite the expression.
Step 2.4
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
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
Simplify each term.
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Step 4.2.3.1
Cancel the common factor of and .
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Step 4.2.3.1.1
Factor out of .
Step 4.2.3.1.2
Factor out of .
Step 4.2.3.1.3
Factor out of .
Step 4.2.3.1.4
Cancel the common factors.
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Step 4.2.3.1.4.1
Factor out of .
Step 4.2.3.1.4.2
Cancel the common factor.
Step 4.2.3.1.4.3
Rewrite the expression.
Step 4.2.3.2
Combine and .
Step 4.2.4
Combine the numerators over the common denominator.
Step 4.2.5
Simplify each term.
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Step 4.2.5.1
Apply the distributive property.
Step 4.2.5.2
Multiply by .
Step 4.2.5.3
Multiply by .
Step 4.2.6
Simplify terms.
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Step 4.2.6.1
Combine the opposite terms in .
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Step 4.2.6.1.1
Add and .
Step 4.2.6.1.2
Add and .
Step 4.2.6.2
Cancel the common factor of .
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Step 4.2.6.2.1
Cancel the common factor.
Step 4.2.6.2.2
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 each term.
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Step 4.3.3.1
Cancel the common factor of .
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Step 4.3.3.1.1
Factor out of .
Step 4.3.3.1.2
Cancel the common factor.
Step 4.3.3.1.3
Rewrite the expression.
Step 4.3.3.2
The functions cosecant and arccosecant are inverses.
Step 4.3.3.3
Apply the distributive property.
Step 4.3.3.4
Cancel the common factor of .
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Step 4.3.3.4.1
Move the leading negative in into the numerator.
Step 4.3.3.4.2
Factor out of .
Step 4.3.3.4.3
Cancel the common factor.
Step 4.3.3.4.4
Rewrite the expression.
Step 4.3.3.5
Multiply by .
Step 4.3.3.6
Multiply by .
Step 4.3.3.7
Cancel the common factor of .
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Step 4.3.3.7.1
Factor out of .
Step 4.3.3.7.2
Cancel the common factor.
Step 4.3.3.7.3
Rewrite the expression.
Step 4.3.4
Combine the opposite terms in .
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Step 4.3.4.1
Add and .
Step 4.3.4.2
Add and .
Step 4.4
Since and , then is the inverse of .