Trigonometry Examples

$f (x) = 2 \sin^{3} (4 x + 1)$

Step 1

Write $f (x) = 2 \sin^{3} (4 x + 1)$ as an equation.

$y = 2 \sin^{3} (4 x + 1)$

Step 2

Interchange the variables.

$x = 2 \sin^{3} (4 y + 1)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $2 \sin^{3} (4 y + 1) = x$ .

$2 \sin^{3} (4 y + 1) = x$

Step 3.2

Divide each term in $2 \sin^{3} (4 y + 1) = x$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 \sin^{3} (4 y + 1) = x$ by $2$ .

$\frac{2 \sin^{3} (4 y + 1)}{2} = \frac{x}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 \sin^{3} (4 y + 1)}{2} = \frac{x}{2}$

Step 3.2.2.1.2

Divide $\sin^{3} (4 y + 1)$ by $1$ .

$\sin^{3} (4 y + 1) = \frac{x}{2}$

Step 3.3

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

$\sin (4 y + 1) = \sqrt[3]{\frac{x}{2}}$

Step 3.4

Simplify $\sqrt[3]{\frac{x}{2}}$ .

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Step 3.4.1

Rewrite $\sqrt[3]{\frac{x}{2}}$ as $\frac{\sqrt[3]{x}}{\sqrt[3]{2}}$ .

$\sin (4 y + 1) = \frac{\sqrt[3]{x}}{\sqrt[3]{2}}$

Step 3.4.2

Multiply $\frac{\sqrt[3]{x}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$\sin (4 y + 1) = \frac{\sqrt[3]{x}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 3.4.3

Combine and simplify the denominator.

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Step 3.4.3.1

Multiply $\frac{\sqrt[3]{x}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$\sin (4 y + 1) = \frac{\sqrt[3]{x} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 3.4.3.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$\sin (4 y + 1) = \frac{\sqrt[3]{x} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 3.4.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (4 y + 1) = \frac{\sqrt[3]{x} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 3.4.3.4

Add $1$ and $2$ .

$\sin (4 y + 1) = \frac{\sqrt[3]{x} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 3.4.3.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 3.4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$\sin (4 y + 1) = \frac{\sqrt[3]{x} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 3.4.3.5.2

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

$\sin (4 y + 1) = \frac{\sqrt[3]{x} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 3.4.3.5.3

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

$\sin (4 y + 1) = \frac{\sqrt[3]{x} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.4.3.5.4

Cancel the common factor of $3$ .

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Step 3.4.3.5.4.1

Cancel the common factor.

$\sin (4 y + 1) = \frac{\sqrt[3]{x} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.4.3.5.4.2

Rewrite the expression.

$\sin (4 y + 1) = \frac{\sqrt[3]{x} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 3.4.3.5.5

Evaluate the exponent.

$\sin (4 y + 1) = \frac{\sqrt[3]{x} {\sqrt[3]{2}}^{2}}{2}$

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$\sin (4 y + 1) = \frac{\sqrt[3]{x} \sqrt[3]{2^{2}}}{2}$

Step 3.4.4.2

Raise $2$ to the power of $2$ .

$\sin (4 y + 1) = \frac{\sqrt[3]{x} \sqrt[3]{4}}{2}$

Step 3.4.5

Simplify with factoring out.

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Step 3.4.5.1

Combine using the product rule for radicals.

$\sin (4 y + 1) = \frac{\sqrt[3]{x \cdot 4}}{2}$

Step 3.4.5.2

Reorder factors in $\frac{\sqrt[3]{x \cdot 4}}{2}$ .

$\sin (4 y + 1) = \frac{\sqrt[3]{4 x}}{2}$

Step 3.5

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

$4 y + 1 = \arcsin (\frac{\sqrt[3]{4 x}}{2})$

Step 3.6

Subtract $1$ from both sides of the equation.

$4 y = \arcsin (\frac{\sqrt[3]{4 x}}{2}) - 1$

Step 3.7

Divide each term in $4 y = \arcsin (\frac{\sqrt[3]{4 x}}{2}) - 1$ by $4$ and simplify.

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Step 3.7.1

Divide each term in $4 y = \arcsin (\frac{\sqrt[3]{4 x}}{2}) - 1$ by $4$ .

$\frac{4 y}{4} = \frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} + \frac{- 1}{4}$

Step 3.7.2

Simplify the left side.

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Step 3.7.2.1

Cancel the common factor of $4$ .

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Step 3.7.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} + \frac{- 1}{4}$

Step 3.7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} + \frac{- 1}{4}$

Step 3.7.3

Simplify the right side.

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Step 3.7.3.1

Move the negative in front of the fraction.

$y = \frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}$

Step 4

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

$f^{- 1} (x) = \frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}$

Step 5

Verify if $f^{- 1} (x) = \frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}$ is the inverse of $f (x) = 2 \sin^{3} (4 x + 1)$ .

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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 $f^{- 1} (2 \sin^{3} (4 x + 1))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 \sin^{3} (4 x + 1)) = \frac{\arcsin (\frac{\sqrt[3]{4 (2 \sin^{3} (4 x + 1))}}{2})}{4} - \frac{1}{4}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Multiply $4$ by $2$ .

$f^{- 1} (2 \sin^{3} (4 x + 1)) = \frac{\arcsin (\frac{\sqrt[3]{8 \sin^{3} (4 x + 1)}}{2})}{4} - \frac{1}{4}$

Step 5.2.3.1.2

Rewrite $8 \sin^{3} (4 x + 1)$ as ${(2 \sin (4 x + 1))}^{3}$ .

$f^{- 1} (2 \sin^{3} (4 x + 1)) = \frac{\arcsin (\frac{\sqrt[3]{{(2 \sin (4 x + 1))}^{3}}}{2})}{4} - \frac{1}{4}$

Step 5.2.3.1.3

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

$f^{- 1} (2 \sin^{3} (4 x + 1)) = \frac{\arcsin (\frac{2 \sin (4 x + 1)}{2})}{4} - \frac{1}{4}$

Step 5.2.3.2

Cancel the common factor of $2$ .

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Step 5.2.3.2.1

Cancel the common factor.

$f^{- 1} (2 \sin^{3} (4 x + 1)) = \frac{\arcsin (\frac{2 \sin (4 x + 1)}{2})}{4} - \frac{1}{4}$

Step 5.2.3.2.2

Divide $\sin (4 x + 1)$ by $1$ .

$f^{- 1} (2 \sin^{3} (4 x + 1)) = \frac{\arcsin (\sin (4 x + 1))}{4} - \frac{1}{4}$

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 $f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 \sin^{3} (4 (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) + 1)$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 \sin^{3} (4 (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4}) + 4 (- \frac{1}{4}) + 1)$

Step 5.3.3.2

Cancel the common factor of $4$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 \sin^{3} (4 (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4}) + 4 (- \frac{1}{4}) + 1)$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 \sin^{3} (\arcsin (\frac{\sqrt[3]{4 x}}{2}) + 4 (- \frac{1}{4}) + 1)$

Step 5.3.3.3

Cancel the common factor of $4$ .

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Step 5.3.3.3.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 \sin^{3} (\arcsin (\frac{\sqrt[3]{4 x}}{2}) + 4 (\frac{- 1}{4}) + 1)$

Step 5.3.3.3.2

Cancel the common factor.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 \sin^{3} (\arcsin (\frac{\sqrt[3]{4 x}}{2}) + 4 (\frac{- 1}{4}) + 1)$

Step 5.3.3.3.3

Rewrite the expression.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 \sin^{3} (\arcsin (\frac{\sqrt[3]{4 x}}{2}) - 1 + 1)$

Step 5.3.4

Combine the opposite terms in $\arcsin (\frac{\sqrt[3]{4 x}}{2}) - 1 + 1$ .

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Step 5.3.4.1

Add $- 1$ and $1$ .

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 \sin^{3} (\arcsin (\frac{\sqrt[3]{4 x}}{2}) + 0)$

Step 5.3.4.2

Add $\arcsin (\frac{\sqrt[3]{4 x}}{2})$ and $0$ .

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 \sin^{3} (\arcsin (\frac{\sqrt[3]{4 x}}{2}))$

Step 5.3.5

The functions sine and arcsine are inverses.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 {(\frac{\sqrt[3]{4 x}}{2})}^{3}$

Step 5.3.6

Apply the product rule to $\frac{\sqrt[3]{4 x}}{2}$ .

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 (\frac{{\sqrt[3]{4 x}}^{3}}{2^{3}})$

Step 5.3.7

Rewrite ${\sqrt[3]{4 x}}^{3}$ as $4 x$ .

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Step 5.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x}$ as ${(4 x)}^{\frac{1}{3}}$ .

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 (\frac{{({(4 x)}^{\frac{1}{3}})}^{3}}{2^{3}})$

Step 5.3.7.2

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

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 (\frac{{(4 x)}^{\frac{1}{3} \cdot 3}}{2^{3}})$

Step 5.3.7.3

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

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 (\frac{{(4 x)}^{\frac{3}{3}}}{2^{3}})$

Step 5.3.7.4

Cancel the common factor of $3$ .

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Step 5.3.7.4.1

Cancel the common factor.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 (\frac{{(4 x)}^{\frac{3}{3}}}{2^{3}})$

Step 5.3.7.4.2

Rewrite the expression.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 (\frac{4 x}{2^{3}})$

Step 5.3.7.5

Simplify.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 (\frac{4 x}{2^{3}})$

Step 5.3.8

Raise $2$ to the power of $3$ .

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 (\frac{4 x}{8})$

Step 5.3.9

Cancel the common factor of $2$ .

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Step 5.3.9.1

Factor $2$ out of $8$ .

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 (\frac{4 x}{2 (4)})$

Step 5.3.9.2

Cancel the common factor.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = 2 (\frac{4 x}{2 \cdot 4})$

Step 5.3.9.3

Rewrite the expression.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = \frac{4 x}{4}$

Step 5.3.10

Cancel the common factor of $4$ .

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Step 5.3.10.1

Cancel the common factor.

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = \frac{4 x}{4}$

Step 5.3.10.2

Divide $x$ by $1$ .

$f (\frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}$ is the inverse of $f (x) = 2 \sin^{3} (4 x + 1)$ .

$f^{- 1} (x) = \frac{\arcsin (\frac{\sqrt[3]{4 x}}{2})}{4} - \frac{1}{4}$