Trigonometry Examples

$7 \sin (8 x)$

Step 1

Interchange the variables.

$x = 7 \sin (8 y)$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $7 \sin (8 y) = x$ .

$7 \sin (8 y) = x$

Step 2.2

Divide each term in $7 \sin (8 y) = x$ by $7$ and simplify.

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Step 2.2.1

Divide each term in $7 \sin (8 y) = x$ by $7$ .

$\frac{7 \sin (8 y)}{7} = \frac{x}{7}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $7$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{7 \sin (8 y)}{7} = \frac{x}{7}$

Step 2.2.2.1.2

Divide $\sin (8 y)$ by $1$ .

$\sin (8 y) = \frac{x}{7}$

Step 2.3

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

$8 y = \arcsin (\frac{x}{7})$

Step 2.4

Divide each term in $8 y = \arcsin (\frac{x}{7})$ by $8$ and simplify.

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Step 2.4.1

Divide each term in $8 y = \arcsin (\frac{x}{7})$ by $8$ .

$\frac{8 y}{8} = \frac{\arcsin (\frac{x}{7})}{8}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $8$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{\arcsin (\frac{x}{7})}{8}$

Step 2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arcsin (\frac{x}{7})}{8}$

Step 3

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

$f^{- 1} (x) = \frac{\arcsin (\frac{x}{7})}{8}$

Step 4

Verify if $f^{- 1} (x) = \frac{\arcsin (\frac{x}{7})}{8}$ is the inverse of $f (x) = 7 \sin (8 x)$ .

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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 $f^{- 1} (7 \sin (8 x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (7 \sin (8 x)) = \frac{\arcsin (\frac{7 \sin (8 x)}{7})}{8}$

Step 4.2.3

Cancel the common factor of $7$ .

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Step 4.2.3.1

Cancel the common factor.

$f^{- 1} (7 \sin (8 x)) = \frac{\arcsin (\frac{7 \sin (8 x)}{7})}{8}$

Step 4.2.3.2

Divide $\sin (8 x)$ by $1$ .

$f^{- 1} (7 \sin (8 x)) = \frac{\arcsin (\sin (8 x))}{8}$

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 $f (\frac{\arcsin (\frac{x}{7})}{8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arcsin (\frac{x}{7})}{8}) = 7 \sin (8 (\frac{\arcsin (\frac{x}{7})}{8}))$

Step 4.3.3

Cancel the common factor of $8$ .

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Step 4.3.3.1

Cancel the common factor.

$f (\frac{\arcsin (\frac{x}{7})}{8}) = 7 \sin (8 (\frac{\arcsin (\frac{x}{7})}{8}))$

Step 4.3.3.2

Rewrite the expression.

$f (\frac{\arcsin (\frac{x}{7})}{8}) = 7 \sin (\arcsin (\frac{x}{7}))$

Step 4.3.4

The functions sine and arcsine are inverses.

$f (\frac{\arcsin (\frac{x}{7})}{8}) = 7 (\frac{x}{7})$

Step 4.3.5

Cancel the common factor of $7$ .

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Step 4.3.5.1

Cancel the common factor.

$f (\frac{\arcsin (\frac{x}{7})}{8}) = 7 (\frac{x}{7})$

Step 4.3.5.2

Rewrite the expression.

$f (\frac{\arcsin (\frac{x}{7})}{8}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arcsin (\frac{x}{7})}{8}$ is the inverse of $f (x) = 7 \sin (8 x)$ .

$f^{- 1} (x) = \frac{\arcsin (\frac{x}{7})}{8}$