Trigonometry Examples

$y = \sin (x + 1)$

Step 1

Interchange the variables.

$x = \sin (y + 1)$

Step 2

Solve for $y$ .

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Step 2.1

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

$\sin (y + 1) = x$

Step 2.2

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

$y + 1 = \arcsin (x)$

Step 2.3

Subtract $1$ from both sides of the equation.

$y = \arcsin (x) - 1$

Step 3

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

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

Step 4

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

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

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

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

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

Step 4.3.3

Combine the opposite terms in $(\arcsin (x) - 1) + 1$ .

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Step 4.3.3.1

Add $- 1$ and $1$ .

$f (\arcsin (x) - 1) = \sin (\arcsin (x) + 0)$

Step 4.3.3.2

Add $\arcsin (x)$ and $0$ .

$f (\arcsin (x) - 1) = \sin (\arcsin (x))$

Step 4.3.4

The functions sine and arcsine are inverses.

$f (\arcsin (x) - 1) = x$

Step 4.4

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

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