Trigonometry Examples

$\sin (- x)$

Step 1

Interchange the variables.

$x = \sin (- y)$

Step 2

Solve for $y$ .

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Step 2.1

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

$\sin (- y) = x$

Step 2.2

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

$- y = \arcsin (x)$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.2.2

Divide $y$ by $1$ .

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

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

$y = - 1 \cdot \arcsin (x)$

Step 2.3.3.2

Rewrite $- 1 \cdot \arcsin (x)$ as $- \arcsin (x)$ .

$y = - \arcsin (x)$

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

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

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

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

Step 4.3.3

Multiply $- (- \arcsin (x))$ .

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Step 4.3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3.2

Multiply $\arcsin (x)$ by $1$ .

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

Step 4.3.4

The functions sine and arcsine are inverses.

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

Step 4.4

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

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