Trigonometry Examples

Find the Inverse f(x)=1/4*sec(pix)
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Solve for .
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Step 3.1
Rewrite the equation as .
Step 3.2
Multiply both sides of the equation by .
Step 3.3
Simplify the left side.
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Step 3.3.1
Simplify .
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Step 3.3.1.1
Combine and .
Step 3.3.1.2
Cancel the common factor of .
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Step 3.3.1.2.1
Cancel the common factor.
Step 3.3.1.2.2
Rewrite the expression.
Step 3.4
Take the inverse secant of both sides of the equation to extract from inside the secant.
Step 3.5
Divide each term in by and simplify.
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Step 3.5.1
Divide each term in by .
Step 3.5.2
Simplify the left side.
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Step 3.5.2.1
Cancel the common factor of .
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Step 3.5.2.1.1
Cancel the common factor.
Step 3.5.2.1.2
Divide by .
Step 4
Replace with to show the final answer.
Step 5
Verify if is the inverse of .
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Step 5.1
To verify the inverse, check if and .
Step 5.2
Evaluate .
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Step 5.2.1
Set up the composite result function.
Step 5.2.2
Evaluate by substituting in the value of into .
Step 5.2.3
Simplify the numerator.
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Step 5.2.3.1
Cancel the common factor of .
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Step 5.2.3.1.1
Cancel the common factor.
Step 5.2.3.1.2
Rewrite the expression.
Step 5.2.3.2
Multiply by .
Step 5.3
Evaluate .
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Step 5.3.1
Set up the composite result function.
Step 5.3.2
Evaluate by substituting in the value of into .
Step 5.3.3
Cancel the common factor of .
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Step 5.3.3.1
Cancel the common factor.
Step 5.3.3.2
Rewrite the expression.
Step 5.3.4
The functions secant and arcsecant are inverses.
Step 5.3.5
Cancel the common factor of .
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Step 5.3.5.1
Factor out of .
Step 5.3.5.2
Cancel the common factor.
Step 5.3.5.3
Rewrite the expression.
Step 5.4
Since and , then is the inverse of .