Trigonometry Examples

$\csc (x - π)$

Step 1

Interchange the variables.

$x = \csc (y - π)$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\csc (y - π) = x$ .

$\csc (y - π) = x$

Step 2.2

Take the inverse cosecant of both sides of the equation to extract $y$ from inside the cosecant.

$y - π = arccsc (x)$

Step 2.3

Add $π$ to both sides of the equation.

$y = arccsc (x) + π$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = arccsc (x) + π$

Step 4

Verify if $f^{- 1} (x) = arccsc (x) + π$ is the inverse of $f (x) = \csc (x - π)$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (\csc (x - π))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\csc (x - π)) = arccsc (\csc (x - π)) + π$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (arccsc (x) + π)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (arccsc (x) + π) = \csc ((arccsc (x) + π) - π)$

Step 4.3.3

Combine the opposite terms in $(arccsc (x) + π) - π$ .

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Step 4.3.3.1

Subtract $π$ from $π$ .

$f (arccsc (x) + π) = \csc (arccsc (x) + 0)$

Step 4.3.3.2

Add $arccsc (x)$ and $0$ .

$f (arccsc (x) + π) = \csc (arccsc (x))$

Step 4.3.4

The functions cosecant and arccosecant are inverses.

$f (arccsc (x) + π) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = arccsc (x) + π$ is the inverse of $f (x) = \csc (x - π)$ .

$f^{- 1} (x) = arccsc (x) + π$