Trigonometry Examples

$f (x) = - 5 \cos (7 x)$

Step 1

Write $f (x) = - 5 \cos (7 x)$ as an equation.

$y = - 5 \cos (7 x)$

Step 2

Interchange the variables.

$x = - 5 \cos (7 y)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 5 \cos (7 y) = x$ .

$- 5 \cos (7 y) = x$

Step 3.2

Divide each term in $- 5 \cos (7 y) = x$ by $- 5$ and simplify.

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Step 3.2.1

Divide each term in $- 5 \cos (7 y) = x$ by $- 5$ .

$\frac{- 5 \cos (7 y)}{- 5} = \frac{x}{- 5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 5 \cos (7 y)}{- 5} = \frac{x}{- 5}$

Step 3.2.2.1.2

Divide $\cos (7 y)$ by $1$ .

$\cos (7 y) = \frac{x}{- 5}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$\cos (7 y) = - \frac{x}{5}$

Step 3.3

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

$7 y = \arccos (- \frac{x}{5})$

Step 3.4

Divide each term in $7 y = \arccos (- \frac{x}{5})$ by $7$ and simplify.

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Step 3.4.1

Divide each term in $7 y = \arccos (- \frac{x}{5})$ by $7$ .

$\frac{7 y}{7} = \frac{\arccos (- \frac{x}{5})}{7}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $7$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{\arccos (- \frac{x}{5})}{7}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arccos (- \frac{x}{5})}{7}$

Step 4

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

$f^{- 1} (x) = \frac{\arccos (- \frac{x}{5})}{7}$

Step 5

Verify if $f^{- 1} (x) = \frac{\arccos (- \frac{x}{5})}{7}$ is the inverse of $f (x) = - 5 \cos (7 x)$ .

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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 $f^{- 1} (- 5 \cos (7 x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 5 \cos (7 x)) = \frac{\arccos (- \frac{- 5 \cos (7 x)}{5})}{7}$

Step 5.2.3

Cancel the common factor of $- 5$ and $5$ .

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Step 5.2.3.1

Factor $5$ out of $- 5 \cos (7 x)$ .

$f^{- 1} (- 5 \cos (7 x)) = \frac{\arccos (- \frac{5 (- \cos (7 x))}{5})}{7}$

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $5$ out of $5$ .

$f^{- 1} (- 5 \cos (7 x)) = \frac{\arccos (- \frac{5 (- \cos (7 x))}{5 (1)})}{7}$

Step 5.2.3.2.2

Cancel the common factor.

$f^{- 1} (- 5 \cos (7 x)) = \frac{\arccos (- \frac{5 (- \cos (7 x))}{5 \cdot 1})}{7}$

Step 5.2.3.2.3

Rewrite the expression.

$f^{- 1} (- 5 \cos (7 x)) = \frac{\arccos (- \frac{- \cos (7 x)}{1})}{7}$

Step 5.2.3.2.4

Divide $- \cos (7 x)$ by $1$ .

$f^{- 1} (- 5 \cos (7 x)) = \frac{\arccos (\cos (7 x))}{7}$

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 $f (\frac{\arccos (- \frac{x}{5})}{7})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arccos (- \frac{x}{5})}{7}) = - 5 \cos (7 (\frac{\arccos (- \frac{x}{5})}{7}))$

Step 5.3.3

Cancel the common factor of $7$ .

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Step 5.3.3.1

Cancel the common factor.

$f (\frac{\arccos (- \frac{x}{5})}{7}) = - 5 \cos (7 (\frac{\arccos (- \frac{x}{5})}{7}))$

Step 5.3.3.2

Rewrite the expression.

$f (\frac{\arccos (- \frac{x}{5})}{7}) = - 5 \cos (\arccos (- \frac{x}{5}))$

Step 5.3.4

The functions cosine and arccosine are inverses.

$f (\frac{\arccos (- \frac{x}{5})}{7}) = - 5 (- \frac{x}{5})$

Step 5.3.5

Cancel the common factor of $5$ .

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Step 5.3.5.1

Move the leading negative in $- \frac{x}{5}$ into the numerator.

$f (\frac{\arccos (- \frac{x}{5})}{7}) = - 5 \frac{- x}{5}$

Step 5.3.5.2

Factor $5$ out of $- 5$ .

$f (\frac{\arccos (- \frac{x}{5})}{7}) = 5 (- 1) (\frac{- x}{5})$

Step 5.3.5.3

Cancel the common factor.

$f (\frac{\arccos (- \frac{x}{5})}{7}) = 5 \cdot (- 1 \frac{- x}{5})$

Step 5.3.5.4

Rewrite the expression.

$f (\frac{\arccos (- \frac{x}{5})}{7}) = - 1 (- x)$

Step 5.3.6

Multiply.

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Step 5.3.6.1

Multiply $- 1$ by $- 1$ .

$f (\frac{\arccos (- \frac{x}{5})}{7}) = 1 x$

Step 5.3.6.2

Multiply $x$ by $1$ .

$f (\frac{\arccos (- \frac{x}{5})}{7}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arccos (- \frac{x}{5})}{7}$ is the inverse of $f (x) = - 5 \cos (7 x)$ .

$f^{- 1} (x) = \frac{\arccos (- \frac{x}{5})}{7}$