Trigonometry Examples

$\sqrt[3]{1 + \tan (x)}$

Step 1

Interchange the variables.

$x = \sqrt[3]{1 + \tan (y)}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sqrt[3]{1 + \tan (y)} = x$ .

$\sqrt[3]{1 + \tan (y)} = x$

Step 2.2

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{1 + \tan (y)}}^{3} = x^{3}$

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{1 + \tan (y)}$ as ${(1 + \tan (y))}^{\frac{1}{3}}$ .

${({(1 + \tan (y))}^{\frac{1}{3}})}^{3} = x^{3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify ${({(1 + \tan (y))}^{\frac{1}{3}})}^{3}$ .

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Step 2.3.2.1.1

Multiply the exponents in ${({(1 + \tan (y))}^{\frac{1}{3}})}^{3}$ .

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Step 2.3.2.1.1.1

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

${(1 + \tan (y))}^{\frac{1}{3} \cdot 3} = x^{3}$

Step 2.3.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

${(1 + \tan (y))}^{\frac{1}{3} \cdot 3} = x^{3}$

Step 2.3.2.1.1.2.2

Rewrite the expression.

${(1 + \tan (y))}^{1} = x^{3}$

Step 2.3.2.1.2

Simplify.

$1 + \tan (y) = x^{3}$

Step 2.4

Subtract $1$ from both sides of the equation.

$\tan (y) = x^{3} - 1$

Step 2.5

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

$y = \arctan (x^{3} - 1)$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = \arctan (x^{3} - 1)$ is the inverse of $f (x) = \sqrt[3]{1 + \tan (x)}$ .

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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 $f^{- 1} (\sqrt[3]{1 + \tan (x)})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[3]{1 + \tan (x)}) = \arctan ({(\sqrt[3]{1 + \tan (x)})}^{3} - 1)$

Step 4.2.3

Rewrite ${\sqrt[3]{1 + \tan (x)}}^{3}$ as $1 + \tan (x)$ .

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Step 4.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{1 + \tan (x)}$ as ${(1 + \tan (x))}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{1 + \tan (x)}) = \arctan ({({(1 + \tan (x))}^{\frac{1}{3}})}^{3} - 1)$

Step 4.2.3.2

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

$f^{- 1} (\sqrt[3]{1 + \tan (x)}) = \arctan ({(1 + \tan (x))}^{\frac{1}{3} \cdot 3} - 1)$

Step 4.2.3.3

Combine $\frac{1}{3}$ and $3$ .

$f^{- 1} (\sqrt[3]{1 + \tan (x)}) = \arctan ({(1 + \tan (x))}^{\frac{3}{3}} - 1)$

Step 4.2.3.4

Cancel the common factor of $3$ .

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Step 4.2.3.4.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{1 + \tan (x)}) = \arctan ({(1 + \tan (x))}^{\frac{3}{3}} - 1)$

Step 4.2.3.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{1 + \tan (x)}) = \arctan ((1 + \tan (x)) - 1)$

Step 4.2.3.5

Simplify.

$f^{- 1} (\sqrt[3]{1 + \tan (x)}) = \arctan (1 + \tan (x) - 1)$

Step 4.2.4

Combine the opposite terms in $1 + \tan (x) - 1$ .

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Step 4.2.4.1

Subtract $1$ from $1$ .

$f^{- 1} (\sqrt[3]{1 + \tan (x)}) = \arctan (0 + \tan (x))$

Step 4.2.4.2

Add $0$ and $\tan (x)$ .

$f^{- 1} (\sqrt[3]{1 + \tan (x)}) = \arctan (\tan (x))$

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

$f (\arctan (x^{3} - 1)) = \sqrt[3]{1 + \tan (\arctan (x^{3} - 1))}$

Step 4.3.3

The functions tangent and arctangent are inverses.

$f (\arctan (x^{3} - 1)) = \sqrt[3]{1 + x^{3} - 1}$

Step 4.3.4

Subtract $1$ from $1$ .

$f (\arctan (x^{3} - 1)) = \sqrt[3]{x^{3} + 0}$

Step 4.3.5

Add $x^{3}$ and $0$ .

$f (\arctan (x^{3} - 1)) = \sqrt[3]{x^{3}}$

Step 4.3.6

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

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

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \arctan (x^{3} - 1)$ is the inverse of $f (x) = \sqrt[3]{1 + \tan (x)}$ .

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