Trigonometry Examples

$f (x) = \cos^{5} (x)$

Step 1

Write $f (x) = \cos^{5} (x)$ as an equation.

$y = \cos^{5} (x)$

Step 2

Interchange the variables.

$x = \cos^{5} (y)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\cos^{5} (y) = x$ .

$\cos^{5} (y) = x$

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (y) = \sqrt[5]{x}$

Step 3.3

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

$y = \arccos (\sqrt[5]{x})$

Step 4

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

$f^{- 1} (x) = \arccos (\sqrt[5]{x})$

Step 5

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

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

$f^{- 1} (\cos^{5} (x)) = \arccos (\sqrt[5]{\cos^{5} (x)})$

Step 5.2.3

Remove parentheses.

$f^{- 1} (\cos^{5} (x)) = \arccos (\sqrt[5]{\cos^{5} (x)})$

Step 5.2.4

Pull terms out from under the radical, assuming real numbers.

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

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

$f (\arccos (\sqrt[5]{x})) = \cos^{5} (\arccos (\sqrt[5]{x}))$

Step 5.3.3

The functions cosine and arccosine are inverses.

$f (\arccos (\sqrt[5]{x})) = {\sqrt[5]{x}}^{5}$

Step 5.3.4

Rewrite ${\sqrt[5]{x}}^{5}$ as $x$ .

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Step 5.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$f (\arccos (\sqrt[5]{x})) = {(x^{\frac{1}{5}})}^{5}$

Step 5.3.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\arccos (\sqrt[5]{x})) = x^{\frac{1}{5} \cdot 5}$

Step 5.3.4.3

Combine $\frac{1}{5}$ and $5$ .

$f (\arccos (\sqrt[5]{x})) = x^{\frac{5}{5}}$

Step 5.3.4.4

Cancel the common factor of $5$ .

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Step 5.3.4.4.1

Cancel the common factor.

$f (\arccos (\sqrt[5]{x})) = x^{\frac{5}{5}}$

Step 5.3.4.4.2

Rewrite the expression.

$f (\arccos (\sqrt[5]{x})) = x$

Step 5.3.4.5

Simplify.

$f (\arccos (\sqrt[5]{x})) = x$

Step 5.4

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

$f^{- 1} (x) = \arccos (\sqrt[5]{x})$