Trigonometry Examples

$f (x) = 7 \arcsin (x + 2)$

Step 1

Write $f (x) = 7 \arcsin (x + 2)$ as an equation.

$y = 7 \arcsin (x + 2)$

Step 2

Interchange the variables.

$x = 7 \arcsin (y + 2)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $7 \arcsin (y + 2) = x$ .

$7 \arcsin (y + 2) = x$

Step 3.2

Divide each term in $7 \arcsin (y + 2) = x$ by $7$ and simplify.

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Step 3.2.1

Divide each term in $7 \arcsin (y + 2) = x$ by $7$ .

$\frac{7 \arcsin (y + 2)}{7} = \frac{x}{7}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $7$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{7 \arcsin (y + 2)}{7} = \frac{x}{7}$

Step 3.2.2.1.2

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

$\arcsin (y + 2) = \frac{x}{7}$

Step 3.3

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

$y + 2 = \sin (\frac{x}{7})$

Step 3.4

Subtract $2$ from both sides of the equation.

$y = \sin (\frac{x}{7}) - 2$

Step 4

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

$f^{- 1} (x) = \sin (\frac{x}{7}) - 2$

Step 5

Verify if $f^{- 1} (x) = \sin (\frac{x}{7}) - 2$ is the inverse of $f (x) = 7 \arcsin (x + 2)$ .

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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 $f^{- 1} (7 \arcsin (x + 2))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (7 \arcsin (x + 2)) = \sin (\frac{7 \arcsin (x + 2)}{7}) - 2$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Cancel the common factor of $7$ .

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Step 5.2.3.1.1

Cancel the common factor.

$f^{- 1} (7 \arcsin (x + 2)) = \sin (\frac{7 \arcsin (x + 2)}{7}) - 2$

Step 5.2.3.1.2

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

$f^{- 1} (7 \arcsin (x + 2)) = \sin (\arcsin (x + 2)) - 2$

Step 5.2.3.2

The functions sine and arcsine are inverses.

$f^{- 1} (7 \arcsin (x + 2)) = x + 2 - 2$

Step 5.2.4

Combine the opposite terms in $x + 2 - 2$ .

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Step 5.2.4.1

Subtract $2$ from $2$ .

$f^{- 1} (7 \arcsin (x + 2)) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (7 \arcsin (x + 2)) = x$

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 $f (\sin (\frac{x}{7}) - 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sin (\frac{x}{7}) - 2) = 7 \arcsin ((\sin (\frac{x}{7}) - 2) + 2)$

Step 5.3.3

Combine the opposite terms in $(\sin (\frac{x}{7}) - 2) + 2$ .

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Step 5.3.3.1

Add $- 2$ and $2$ .

$f (\sin (\frac{x}{7}) - 2) = 7 \arcsin (\sin (\frac{x}{7}) + 0)$

Step 5.3.3.2

Add $\sin (\frac{x}{7})$ and $0$ .

$f (\sin (\frac{x}{7}) - 2) = 7 \arcsin (\sin (\frac{x}{7}))$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sin (\frac{x}{7}) - 2$ is the inverse of $f (x) = 7 \arcsin (x + 2)$ .

$f^{- 1} (x) = \sin (\frac{x}{7}) - 2$