Trigonometry Examples

$y = - \frac{1}{6} \sin (\frac{x}{4})$

Step 1

Interchange the variables.

$x = - \frac{1}{6} \cdot \sin (\frac{y}{4})$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- \frac{1}{6} \cdot \sin (\frac{y}{4}) = x$ .

$- \frac{1}{6} \cdot \sin (\frac{y}{4}) = x$

Step 2.2

Multiply both sides of the equation by $- 6$ .

$- 6 (- \frac{1}{6} \sin (\frac{y}{4})) = - 6 x$

Step 2.3

Simplify the left side.

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Step 2.3.1

Simplify $- 6 (- \frac{1}{6} \sin (\frac{y}{4}))$ .

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Step 2.3.1.1

Combine $\sin (\frac{y}{4})$ and $\frac{1}{6}$ .

$- 6 (- \frac{\sin (\frac{y}{4})}{6}) = - 6 x$

Step 2.3.1.2

Cancel the common factor of $6$ .

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Step 2.3.1.2.1

Move the leading negative in $- \frac{\sin (\frac{y}{4})}{6}$ into the numerator.

$- 6 \frac{- \sin (\frac{y}{4})}{6} = - 6 x$

Step 2.3.1.2.2

Factor $6$ out of $- 6$ .

$6 (- 1) \frac{- \sin (\frac{y}{4})}{6} = - 6 x$

Step 2.3.1.2.3

Cancel the common factor.

$6 \cdot - 1 \frac{- \sin (\frac{y}{4})}{6} = - 6 x$

Step 2.3.1.2.4

Rewrite the expression.

$- - \sin (\frac{y}{4}) = - 6 x$

Step 2.3.1.3

Multiply.

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Step 2.3.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (\frac{y}{4}) = - 6 x$

Step 2.3.1.3.2

Multiply $\sin (\frac{y}{4})$ by $1$ .

$\sin (\frac{y}{4}) = - 6 x$

Step 2.4

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

$\frac{y}{4} = \arcsin (- 6 x)$

Step 2.5

Multiply both sides of the equation by $4$ .

$4 \frac{y}{4} = 4 \arcsin (- 6 x)$

Step 2.6

Simplify the left side.

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Step 2.6.1

Cancel the common factor of $4$ .

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Step 2.6.1.1

Cancel the common factor.

$4 \frac{y}{4} = 4 \arcsin (- 6 x)$

Step 2.6.1.2

Rewrite the expression.

$y = 4 \arcsin (- 6 x)$

Step 3

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

$f^{- 1} (x) = 4 \arcsin (- 6 x)$

Step 4

Verify if $f^{- 1} (x) = 4 \arcsin (- 6 x)$ is the inverse of $f (x) = - \frac{1}{6} \cdot \sin (\frac{x}{4})$ .

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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 $f^{- 1} (- \frac{1}{6} \cdot \sin (\frac{x}{4}))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{1}{6} \cdot \sin (\frac{x}{4})) = 4 \arcsin (- 6 (- \frac{1}{6} \cdot \sin (\frac{x}{4})))$

Step 4.2.3

Combine $\sin (\frac{x}{4})$ and $\frac{1}{6}$ .

$f^{- 1} (- \frac{1}{6} \cdot \sin (\frac{x}{4})) = 4 \arcsin (- 6 (- \frac{\sin (\frac{x}{4})}{6}))$

Step 4.2.4

Cancel the common factor of $6$ .

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Step 4.2.4.1

Move the leading negative in $- \frac{\sin (\frac{x}{4})}{6}$ into the numerator.

$f^{- 1} (- \frac{1}{6} \cdot \sin (\frac{x}{4})) = 4 \arcsin (- 6 \frac{- \sin (\frac{x}{4})}{6})$

Step 4.2.4.2

Factor $6$ out of $- 6$ .

$f^{- 1} (- \frac{1}{6} \cdot \sin (\frac{x}{4})) = 4 \arcsin (6 (- 1) (\frac{- \sin (\frac{x}{4})}{6}))$

Step 4.2.4.3

Cancel the common factor.

$f^{- 1} (- \frac{1}{6} \cdot \sin (\frac{x}{4})) = 4 \arcsin (6 \cdot (- 1 \frac{- \sin (\frac{x}{4})}{6}))$

Step 4.2.4.4

Rewrite the expression.

$f^{- 1} (- \frac{1}{6} \cdot \sin (\frac{x}{4})) = 4 \arcsin (- 1 (- \sin (\frac{x}{4})))$

Step 4.2.5

Multiply.

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Step 4.2.5.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \frac{1}{6} \cdot \sin (\frac{x}{4})) = 4 \arcsin (1 \sin (\frac{x}{4}))$

Step 4.2.5.2

Multiply $\sin (\frac{x}{4})$ by $1$ .

$f^{- 1} (- \frac{1}{6} \cdot \sin (\frac{x}{4})) = 4 \arcsin (\sin (\frac{x}{4}))$

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 $f (4 \arcsin (- 6 x))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (4 \arcsin (- 6 x)) = - \frac{1}{6} \cdot \sin (\frac{4 \arcsin (- 6 x)}{4})$

Step 4.3.3

Cancel the common factor of $4$ .

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Step 4.3.3.1

Cancel the common factor.

$f (4 \arcsin (- 6 x)) = - \frac{1}{6} \cdot \sin (\frac{4 \arcsin (- 6 x)}{4})$

Step 4.3.3.2

Divide $\arcsin (- 6 x)$ by $1$ .

$f (4 \arcsin (- 6 x)) = - \frac{1}{6} \cdot \sin (\arcsin (- 6 x))$

Step 4.3.4

The functions sine and arcsine are inverses.

$f (4 \arcsin (- 6 x)) = - \frac{1}{6} \cdot (- 6 x)$

Step 4.3.5

Cancel the common factor of $6$ .

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Step 4.3.5.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$f (4 \arcsin (- 6 x)) = \frac{- 1}{6} \cdot (- 6 x)$

Step 4.3.5.2

Factor $6$ out of $- 6 x$ .

$f (4 \arcsin (- 6 x)) = \frac{- 1}{6} \cdot (6 (- x))$

Step 4.3.5.3

Cancel the common factor.

$f (4 \arcsin (- 6 x)) = \frac{- 1}{6} \cdot (6 (- x))$

Step 4.3.5.4

Rewrite the expression.

$f (4 \arcsin (- 6 x)) = - 1 \cdot (- x)$

Step 4.3.6

Multiply.

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Step 4.3.6.1

Multiply $- 1$ by $- 1$ .

$f (4 \arcsin (- 6 x)) = 1 \cdot x$

Step 4.3.6.2

Multiply $x$ by $1$ .

$f (4 \arcsin (- 6 x)) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 4 \arcsin (- 6 x)$ is the inverse of $f (x) = - \frac{1}{6} \cdot \sin (\frac{x}{4})$ .

$f^{- 1} (x) = 4 \arcsin (- 6 x)$