Trigonometry Examples

$f (x) = 3 + \cos (\frac{x}{4})$

Step 1

Write $f (x) = 3 + \cos (\frac{x}{4})$ as an equation.

$y = 3 + \cos (\frac{x}{4})$

Step 2

Interchange the variables.

$x = 3 + \cos (\frac{y}{4})$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 + \cos (\frac{y}{4}) = x$ .

$3 + \cos (\frac{y}{4}) = x$

Step 3.2

Subtract $3$ from both sides of the equation.

$\cos (\frac{y}{4}) = x - 3$

Step 3.3

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

$\frac{y}{4} = \arccos (x - 3)$

Step 3.4

Multiply both sides of the equation by $4$ .

$4 \frac{y}{4} = 4 \arccos (x - 3)$

Step 3.5

Simplify the left side.

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Step 3.5.1

Cancel the common factor of $4$ .

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Step 3.5.1.1

Cancel the common factor.

$4 \frac{y}{4} = 4 \arccos (x - 3)$

Step 3.5.1.2

Rewrite the expression.

$y = 4 \arccos (x - 3)$

Step 4

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

$f^{- 1} (x) = 4 \arccos (x - 3)$

Step 5

Verify if $f^{- 1} (x) = 4 \arccos (x - 3)$ is the inverse of $f (x) = 3 + \cos (\frac{x}{4})$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (3 + \cos (\frac{x}{4}))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (3 + \cos (\frac{x}{4})) = 4 \arccos ((3 + \cos (\frac{x}{4})) - 3)$

Step 5.2.3

Combine the opposite terms in $(3 + \cos (\frac{x}{4})) - 3$ .

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Step 5.2.3.1

Subtract $3$ from $3$ .

$f^{- 1} (3 + \cos (\frac{x}{4})) = 4 \arccos (0 + \cos (\frac{x}{4}))$

Step 5.2.3.2

Add $0$ and $\cos (\frac{x}{4})$ .

$f^{- 1} (3 + \cos (\frac{x}{4})) = 4 \arccos (\cos (\frac{x}{4}))$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (4 \arccos (x - 3))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (4 \arccos (x - 3)) = 3 + \cos (\frac{4 \arccos (x - 3)}{4})$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Cancel the common factor of $4$ .

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Step 5.3.3.1.1

Cancel the common factor.

$f (4 \arccos (x - 3)) = 3 + \cos (\frac{4 \arccos (x - 3)}{4})$

Step 5.3.3.1.2

Divide $\arccos (x - 3)$ by $1$ .

$f (4 \arccos (x - 3)) = 3 + \cos (\arccos (x - 3))$

Step 5.3.3.2

The functions cosine and arccosine are inverses.

$f (4 \arccos (x - 3)) = 3 + x - 3$

Step 5.3.4

Combine the opposite terms in $3 + x - 3$ .

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Step 5.3.4.1

Subtract $3$ from $3$ .

$f (4 \arccos (x - 3)) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (4 \arccos (x - 3)) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 4 \arccos (x - 3)$ is the inverse of $f (x) = 3 + \cos (\frac{x}{4})$ .

$f^{- 1} (x) = 4 \arccos (x - 3)$