Trigonometry Examples

$f (x) = 5 \sin (x + 2)$

Step 1

Write $f (x) = 5 \sin (x + 2)$ as an equation.

$y = 5 \sin (x + 2)$

Step 2

Interchange the variables.

$x = 5 \sin (y + 2)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $5 \sin (y + 2) = x$ .

$5 \sin (y + 2) = x$

Step 3.2

Divide each term in $5 \sin (y + 2) = x$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 \sin (y + 2) = x$ by $5$ .

$\frac{5 \sin (y + 2)}{5} = \frac{x}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 \sin (y + 2)}{5} = \frac{x}{5}$

Step 3.2.2.1.2

Divide $\sin (y + 2)$ by $1$ .

$\sin (y + 2) = \frac{x}{5}$

Step 3.3

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

$y + 2 = \arcsin (\frac{x}{5})$

Step 3.4

Subtract $2$ from both sides of the equation.

$y = \arcsin (\frac{x}{5}) - 2$

Step 4

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

$f^{- 1} (x) = \arcsin (\frac{x}{5}) - 2$

Step 5

Verify if $f^{- 1} (x) = \arcsin (\frac{x}{5}) - 2$ is the inverse of $f (x) = 5 \sin (x + 2)$ .

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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 $f^{- 1} (5 \sin (x + 2))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 \sin (x + 2)) = \arcsin (\frac{5 \sin (x + 2)}{5}) - 2$

Step 5.2.3

Cancel the common factor of $5$ .

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Step 5.2.3.1

Cancel the common factor.

$f^{- 1} (5 \sin (x + 2)) = \arcsin (\frac{5 \sin (x + 2)}{5}) - 2$

Step 5.2.3.2

Divide $\sin (x + 2)$ by $1$ .

$f^{- 1} (5 \sin (x + 2)) = \arcsin (\sin (x + 2)) - 2$

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 $f (\arcsin (\frac{x}{5}) - 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\arcsin (\frac{x}{5}) - 2) = 5 \sin ((\arcsin (\frac{x}{5}) - 2) + 2)$

Step 5.3.3

Combine the opposite terms in $(\arcsin (\frac{x}{5}) - 2) + 2$ .

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Step 5.3.3.1

Add $- 2$ and $2$ .

$f (\arcsin (\frac{x}{5}) - 2) = 5 \sin (\arcsin (\frac{x}{5}) + 0)$

Step 5.3.3.2

Add $\arcsin (\frac{x}{5})$ and $0$ .

$f (\arcsin (\frac{x}{5}) - 2) = 5 \sin (\arcsin (\frac{x}{5}))$

Step 5.3.4

The functions sine and arcsine are inverses.

$f (\arcsin (\frac{x}{5}) - 2) = 5 (\frac{x}{5})$

Step 5.3.5

Cancel the common factor of $5$ .

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Step 5.3.5.1

Cancel the common factor.

$f (\arcsin (\frac{x}{5}) - 2) = 5 (\frac{x}{5})$

Step 5.3.5.2

Rewrite the expression.

$f (\arcsin (\frac{x}{5}) - 2) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \arcsin (\frac{x}{5}) - 2$ is the inverse of $f (x) = 5 \sin (x + 2)$ .

$f^{- 1} (x) = \arcsin (\frac{x}{5}) - 2$