Trigonometry Examples

$f (x) = 2 \sin (- x + 1)$

Step 1

Write $f (x) = 2 \sin (- x + 1)$ as an equation.

$y = 2 \sin (- x + 1)$

Step 2

Interchange the variables.

$x = 2 \sin (- y + 1)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $2 \sin (- y + 1) = x$ .

$2 \sin (- y + 1) = x$

Step 3.2

Divide each term in $2 \sin (- y + 1) = x$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 \sin (- y + 1) = x$ by $2$ .

$\frac{2 \sin (- y + 1)}{2} = \frac{x}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 \sin (- y + 1)}{2} = \frac{x}{2}$

Step 3.2.2.1.2

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

$\sin (- y + 1) = \frac{x}{2}$

Step 3.3

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

$- y + 1 = \arcsin (\frac{x}{2})$

Step 3.4

Subtract $1$ from both sides of the equation.

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

Step 3.5

Divide each term in $- y = \arcsin (\frac{x}{2}) - 1$ by $- 1$ and simplify.

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Step 3.5.1

Divide each term in $- y = \arcsin (\frac{x}{2}) - 1$ by $- 1$ .

$\frac{- y}{- 1} = \frac{\arcsin (\frac{x}{2})}{- 1} + \frac{- 1}{- 1}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{\arcsin (\frac{x}{2})}{- 1} + \frac{- 1}{- 1}$

Step 3.5.2.2

Divide $y$ by $1$ .

$y = \frac{\arcsin (\frac{x}{2})}{- 1} + \frac{- 1}{- 1}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Simplify each term.

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Step 3.5.3.1.1

Move the negative one from the denominator of $\frac{\arcsin (\frac{x}{2})}{- 1}$ .

$y = - 1 \cdot \arcsin (\frac{x}{2}) + \frac{- 1}{- 1}$

Step 3.5.3.1.2

Rewrite $- 1 \cdot \arcsin (\frac{x}{2})$ as $- \arcsin (\frac{x}{2})$ .

$y = - \arcsin (\frac{x}{2}) + \frac{- 1}{- 1}$

Step 3.5.3.1.3

Divide $- 1$ by $- 1$ .

$y = - \arcsin (\frac{x}{2}) + 1$

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Cancel the common factor of $2$ .

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Step 5.2.3.1

Cancel the common factor.

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

Step 5.2.3.2

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

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Multiply $- - \arcsin (\frac{x}{2})$ .

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Step 5.3.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.3.2.2

Multiply $\arcsin (\frac{x}{2})$ by $1$ .

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

Step 5.3.3.3

Multiply $- 1$ by $1$ .

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

Step 5.3.4

Combine the opposite terms in $\arcsin (\frac{x}{2}) - 1 + 1$ .

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Step 5.3.4.1

Add $- 1$ and $1$ .

$f (- \arcsin (\frac{x}{2}) + 1) = 2 \sin (\arcsin (\frac{x}{2}) + 0)$

Step 5.3.4.2

Add $\arcsin (\frac{x}{2})$ and $0$ .

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

Step 5.3.5

The functions sine and arcsine are inverses.

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

Step 5.3.6

Cancel the common factor of $2$ .

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Step 5.3.6.1

Cancel the common factor.

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

Step 5.3.6.2

Rewrite the expression.

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

Step 5.4

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

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