Trigonometry Examples

$f (x) = \sin (h (x))$

Step 1

Write $f (x) = \sin (h (x))$ as an equation.

$y = \sin (h (x))$

Step 2

Interchange the variables.

$h = \sin (y (x))$

Step 3

Solve for $y$ .

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Step 3.1

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

$\sin (y x) = h$

Step 3.2

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

$y x = \arcsin (h)$

Step 3.3

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

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Step 3.3.1

Divide each term in $y x = \arcsin (h)$ by $x$ .

$\frac{y x}{x} = \frac{\arcsin (h)}{x}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $x$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{y x}{x} = \frac{\arcsin (h)}{x}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arcsin (h)}{x}$

Step 4

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

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

Step 5

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

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (h)) = h$ and $f (f^{- 1} (h)) = h$ .

Step 5.2

Evaluate $f^{- 1} (f (h))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (h))$

Step 5.2.2

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

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

Step 5.2.3

Multiply $h$ by $x$ .

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

Step 5.3

Evaluate $f (f^{- 1} (h))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (h))$

Step 5.3.2

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

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

Step 5.3.3

Cancel the common factor of $x$ .

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Step 5.3.3.1

Cancel the common factor.

$f (\frac{\arcsin (h)}{x}) = \sin (\frac{\arcsin (h)}{x} \cdot x)$

Step 5.3.3.2

Rewrite the expression.

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

Step 5.3.4

The functions sine and arcsine are inverses.

$f (\frac{\arcsin (h)}{x}) = h$

Step 5.4

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

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