Trigonometry Examples

$- 4 \csc (\frac{1}{2} x) + 2$

Step 1

Interchange the variables.

$x = - 4 \csc (\frac{1}{2} y) + 2$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- 4 \csc (\frac{1}{2} y) + 2 = x$ .

$- 4 \csc (\frac{1}{2} y) + 2 = x$

Step 2.2

Subtract $2$ from both sides of the equation.

$- 4 \csc (\frac{1}{2} y) = x - 2$

Step 2.3

Divide each term in $- 4 \csc (\frac{1}{2} y) = x - 2$ by $- 4$ and simplify.

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Step 2.3.1

Divide each term in $- 4 \csc (\frac{1}{2} y) = x - 2$ by $- 4$ .

$\frac{- 4 \csc (\frac{1}{2} y)}{- 4} = \frac{x}{- 4} + \frac{- 2}{- 4}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 4$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 4 \csc (\frac{1}{2} y)}{- 4} = \frac{x}{- 4} + \frac{- 2}{- 4}$

Step 2.3.2.1.2

Divide $\csc (\frac{1}{2} y)$ by $1$ .

$\csc (\frac{1}{2} y) = \frac{x}{- 4} + \frac{- 2}{- 4}$

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.

$\csc (\frac{1}{2} y) = - \frac{x}{4} + \frac{- 2}{- 4}$

Step 2.3.3.1.2

Cancel the common factor of $- 2$ and $- 4$ .

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Step 2.3.3.1.2.1

Factor $- 2$ out of $- 2$ .

$\csc (\frac{1}{2} y) = - \frac{x}{4} + \frac{- 2 (1)}{- 4}$

Step 2.3.3.1.2.2

Cancel the common factors.

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Step 2.3.3.1.2.2.1

Factor $- 2$ out of $- 4$ .

$\csc (\frac{1}{2} y) = - \frac{x}{4} + \frac{- 2 \cdot 1}{- 2 \cdot 2}$

Step 2.3.3.1.2.2.2

Cancel the common factor.

$\csc (\frac{1}{2} y) = - \frac{x}{4} + \frac{- 2 \cdot 1}{- 2 \cdot 2}$

Step 2.3.3.1.2.2.3

Rewrite the expression.

$\csc (\frac{1}{2} y) = - \frac{x}{4} + \frac{1}{2}$

Step 2.4

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

$\frac{1}{2} y = arccsc (- \frac{x}{4} + \frac{1}{2})$

Step 2.5

Simplify the left side.

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Step 2.5.1

Combine $\frac{1}{2}$ and $y$ .

$\frac{y}{2} = arccsc (- \frac{x}{4} + \frac{1}{2})$

Step 2.6

Multiply both sides of the equation by $2$ .

$2 \frac{y}{2} = 2 arccsc (- \frac{x}{4} + \frac{1}{2})$

Step 2.7

Simplify the left side.

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Step 2.7.1

Cancel the common factor of $2$ .

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Step 2.7.1.1

Cancel the common factor.

$2 \frac{y}{2} = 2 arccsc (- \frac{x}{4} + \frac{1}{2})$

Step 2.7.1.2

Rewrite the expression.

$y = 2 arccsc (- \frac{x}{4} + \frac{1}{2})$

Step 3

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

$f^{- 1} (x) = 2 arccsc (- \frac{x}{4} + \frac{1}{2})$

Step 4

Verify if $f^{- 1} (x) = 2 arccsc (- \frac{x}{4} + \frac{1}{2})$ is the inverse of $f (x) = - 4 \csc (\frac{1}{2} x) + 2$ .

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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 $f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (- \frac{- 4 \csc (\frac{1}{2} x) + 2}{4} + \frac{1}{2})$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Cancel the common factor of $- 4 \csc (\frac{1}{2} x) + 2$ and $4$ .

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Step 4.2.3.1.1

Factor $2$ out of $- 4 \csc (\frac{1}{2} x)$ .

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (- \frac{2 (- 2 \csc (\frac{1}{2} x)) + 2}{4} + \frac{1}{2})$

Step 4.2.3.1.2

Factor $2$ out of $2$ .

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (- \frac{2 (- 2 \csc (\frac{1}{2} x)) + 2 (1)}{4} + \frac{1}{2})$

Step 4.2.3.1.3

Factor $2$ out of $2 (- 2 \csc (\frac{1}{2} x)) + 2 (1)$ .

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (- \frac{2 (- 2 \csc (\frac{1}{2} x) + 1)}{4} + \frac{1}{2})$

Step 4.2.3.1.4

Cancel the common factors.

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Step 4.2.3.1.4.1

Factor $2$ out of $4$ .

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (- \frac{2 (- 2 \csc (\frac{1}{2} x) + 1)}{2 (2)} + \frac{1}{2})$

Step 4.2.3.1.4.2

Cancel the common factor.

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (- \frac{2 (- 2 \csc (\frac{1}{2} x) + 1)}{2 \cdot 2} + \frac{1}{2})$

Step 4.2.3.1.4.3

Rewrite the expression.

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (- \frac{- 2 \csc (\frac{1}{2} x) + 1}{2} + \frac{1}{2})$

Step 4.2.3.2

Combine $\frac{1}{2}$ and $x$ .

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (- \frac{- 2 \csc (\frac{x}{2}) + 1}{2} + \frac{1}{2})$

Step 4.2.4

Combine the numerators over the common denominator.

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (\frac{- (- 2 \csc (\frac{x}{2}) + 1) + 1}{2})$

Step 4.2.5

Simplify each term.

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Step 4.2.5.1

Apply the distributive property.

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (\frac{- (- 2 \csc (\frac{x}{2})) - 1 \cdot 1 + 1}{2})$

Step 4.2.5.2

Multiply $- 2$ by $- 1$ .

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (\frac{2 \csc (\frac{x}{2}) - 1 \cdot 1 + 1}{2})$

Step 4.2.5.3

Multiply $- 1$ by $1$ .

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (\frac{2 \csc (\frac{x}{2}) - 1 + 1}{2})$

Step 4.2.6

Simplify terms.

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Step 4.2.6.1

Combine the opposite terms in $2 \csc (\frac{x}{2}) - 1 + 1$ .

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Step 4.2.6.1.1

Add $- 1$ and $1$ .

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (\frac{2 \csc (\frac{x}{2}) + 0}{2})$

Step 4.2.6.1.2

Add $2 \csc (\frac{x}{2})$ and $0$ .

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (\frac{2 \csc (\frac{x}{2})}{2})$

Step 4.2.6.2

Cancel the common factor of $2$ .

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Step 4.2.6.2.1

Cancel the common factor.

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (\frac{2 \csc (\frac{x}{2})}{2})$

Step 4.2.6.2.2

Divide $\csc (\frac{x}{2})$ by $1$ .

$f^{- 1} (- 4 \csc (\frac{1}{2} x) + 2) = 2 arccsc (\csc (\frac{x}{2}))$

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 $f (2 arccsc (- \frac{x}{4} + \frac{1}{2}))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = - 4 \csc (\frac{1}{2} \cdot (2 arccsc (- \frac{x}{4} + \frac{1}{2}))) + 2$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Cancel the common factor of $2$ .

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Step 4.3.3.1.1

Factor $2$ out of $2 arccsc (- \frac{x}{4} + \frac{1}{2})$ .

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = - 4 \csc (\frac{1}{2} \cdot (2 (arccsc (- \frac{x}{4} + \frac{1}{2})))) + 2$

Step 4.3.3.1.2

Cancel the common factor.

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = - 4 \csc (\frac{1}{2} \cdot (2 arccsc (- \frac{x}{4} + \frac{1}{2}))) + 2$

Step 4.3.3.1.3

Rewrite the expression.

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = - 4 \csc (arccsc (- \frac{x}{4} + \frac{1}{2})) + 2$

Step 4.3.3.2

The functions cosecant and arccosecant are inverses.

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = - 4 (- \frac{x}{4} + \frac{1}{2}) + 2$

Step 4.3.3.3

Apply the distributive property.

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = - 4 (- \frac{x}{4}) - 4 (\frac{1}{2}) + 2$

Step 4.3.3.4

Cancel the common factor of $4$ .

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Step 4.3.3.4.1

Move the leading negative in $- \frac{x}{4}$ into the numerator.

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = - 4 \frac{- x}{4} - 4 (\frac{1}{2}) + 2$

Step 4.3.3.4.2

Factor $4$ out of $- 4$ .

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = 4 (- 1) (\frac{- x}{4}) - 4 (\frac{1}{2}) + 2$

Step 4.3.3.4.3

Cancel the common factor.

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = 4 \cdot (- 1 \frac{- x}{4}) - 4 (\frac{1}{2}) + 2$

Step 4.3.3.4.4

Rewrite the expression.

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = - 1 (- x) - 4 (\frac{1}{2}) + 2$

Step 4.3.3.5

Multiply $- 1$ by $- 1$ .

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = 1 x - 4 (\frac{1}{2}) + 2$

Step 4.3.3.6

Multiply $x$ by $1$ .

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = x - 4 (\frac{1}{2}) + 2$

Step 4.3.3.7

Cancel the common factor of $2$ .

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Step 4.3.3.7.1

Factor $2$ out of $- 4$ .

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = x + 2 (- 2) (\frac{1}{2}) + 2$

Step 4.3.3.7.2

Cancel the common factor.

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = x + 2 \cdot (- 2 (\frac{1}{2})) + 2$

Step 4.3.3.7.3

Rewrite the expression.

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = x - 2 + 2$

Step 4.3.4

Combine the opposite terms in $x - 2 + 2$ .

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Step 4.3.4.1

Add $- 2$ and $2$ .

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (2 arccsc (- \frac{x}{4} + \frac{1}{2})) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 2 arccsc (- \frac{x}{4} + \frac{1}{2})$ is the inverse of $f (x) = - 4 \csc (\frac{1}{2} x) + 2$ .

$f^{- 1} (x) = 2 arccsc (- \frac{x}{4} + \frac{1}{2})$