Trigonometry Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 7$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $\cos (5 y)$ by $1$ .

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

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.

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $5$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Divide $y$ by $1$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Cancel the common factor of $- 7$ and $7$ .

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Step 5.2.3.1

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

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

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $7$ out of $7$ .

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

Step 5.2.3.2.2

Cancel the common factor.

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

Step 5.2.3.2.3

Rewrite the expression.

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

Step 5.2.3.2.4

Divide $- \cos (5 x)$ by $1$ .

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

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

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

Step 5.3.3

Cancel the common factor of $5$ .

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Step 5.3.3.1

Cancel the common factor.

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

Step 5.3.3.2

Rewrite the expression.

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

Step 5.3.4

The functions cosine and arccosine are inverses.

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

Step 5.3.5

Cancel the common factor of $7$ .

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Step 5.3.5.1

Move the leading negative in $- \frac{x}{7}$ into the numerator.

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

Step 5.3.5.2

Factor $7$ out of $- 7$ .

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

Step 5.3.5.3

Cancel the common factor.

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

Step 5.3.5.4

Rewrite the expression.

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

Step 5.3.6

Multiply.

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Step 5.3.6.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.6.2

Multiply $x$ by $1$ .

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

Step 5.4

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

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