Trigonometry Examples

$y = \csc (3 x) - 1$

Step 1

Interchange the variables.

$x = \csc (3 y) - 1$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\csc (3 y) - 1 = x$ .

$\csc (3 y) - 1 = x$

Step 2.2

Add $1$ to both sides of the equation.

$\csc (3 y) = x + 1$

Step 2.3

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

$3 y = arccsc (x + 1)$

Step 2.4

Divide each term in $3 y = arccsc (x + 1)$ by $3$ and simplify.

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Step 2.4.1

Divide each term in $3 y = arccsc (x + 1)$ by $3$ .

$\frac{3 y}{3} = \frac{arccsc (x + 1)}{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{arccsc (x + 1)}{3}$

Step 2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{arccsc (x + 1)}{3}$

Step 3

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

$f^{- 1} (x) = \frac{arccsc (x + 1)}{3}$

Step 4

Verify if $f^{- 1} (x) = \frac{arccsc (x + 1)}{3}$ is the inverse of $f (x) = \csc (3 x) - 1$ .

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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 $f^{- 1} (\csc (3 x) - 1)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\csc (3 x) - 1) = \frac{arccsc ((\csc (3 x) - 1) + 1)}{3}$

Step 4.2.3

Combine the opposite terms in $\csc (3 x) - 1 + 1$ .

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Step 4.2.3.1

Add $- 1$ and $1$ .

$f^{- 1} (\csc (3 x) - 1) = \frac{arccsc (\csc (3 x) + 0)}{3}$

Step 4.2.3.2

Add $\csc (3 x)$ and $0$ .

$f^{- 1} (\csc (3 x) - 1) = \frac{arccsc (\csc (3 x))}{3}$

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 $f (\frac{arccsc (x + 1)}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{arccsc (x + 1)}{3}) = \csc (3 (\frac{arccsc (x + 1)}{3})) - 1$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Cancel the common factor of $3$ .

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Step 4.3.3.1.1

Cancel the common factor.

$f (\frac{arccsc (x + 1)}{3}) = \csc (3 (\frac{arccsc (x + 1)}{3})) - 1$

Step 4.3.3.1.2

Rewrite the expression.

$f (\frac{arccsc (x + 1)}{3}) = \csc (arccsc (x + 1)) - 1$

Step 4.3.3.2

The functions cosecant and arccosecant are inverses.

$f (\frac{arccsc (x + 1)}{3}) = x + 1 - 1$

Step 4.3.4

Combine the opposite terms in $x + 1 - 1$ .

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Step 4.3.4.1

Subtract $1$ from $1$ .

$f (\frac{arccsc (x + 1)}{3}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\frac{arccsc (x + 1)}{3}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{arccsc (x + 1)}{3}$ is the inverse of $f (x) = \csc (3 x) - 1$ .

$f^{- 1} (x) = \frac{arccsc (x + 1)}{3}$