Trigonometry Examples

$y = 2 \sin (x)$

Step 1

Interchange the variables.

$x = 2 \sin (y)$

Step 2

Solve for $y$ .

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Step 2.1

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

$2 \sin (y) = x$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

Cancel the common factor.

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

Step 4.2.3.2

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

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

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

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

Step 4.3.3

The functions sine and arcsine are inverses.

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

Step 4.3.4

Cancel the common factor of $2$ .

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Step 4.3.4.1

Cancel the common factor.

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

Step 4.3.4.2

Rewrite the expression.

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

Step 4.4

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

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