Trigonometry Examples

$- \sin (5 x) - 8$

Step 1

Interchange the variables.

$x = - \sin (5 y) - 8$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- \sin (5 y) - 8 = x$ .

$- \sin (5 y) - 8 = x$

Step 2.2

Add $8$ to both sides of the equation.

$- \sin (5 y) = x + 8$

Step 2.3

Divide each term in $- \sin (5 y) = x + 8$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- \sin (5 y) = x + 8$ by $- 1$ .

$\frac{- \sin (5 y)}{- 1} = \frac{x}{- 1} + \frac{8}{- 1}$

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.

$\frac{\sin (5 y)}{1} = \frac{x}{- 1} + \frac{8}{- 1}$

Step 2.3.2.2

Divide $\sin (5 y)$ by $1$ .

$\sin (5 y) = \frac{x}{- 1} + \frac{8}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

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

$\sin (5 y) = - 1 \cdot x + \frac{8}{- 1}$

Step 2.3.3.1.2

Rewrite $- 1 \cdot x$ as $- x$ .

$\sin (5 y) = - x + \frac{8}{- 1}$

Step 2.3.3.1.3

Divide $8$ by $- 1$ .

$\sin (5 y) = - x - 8$

Step 2.4

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

$5 y = \arcsin (- x - 8)$

Step 2.5

Divide each term in $5 y = \arcsin (- x - 8)$ by $5$ and simplify.

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Step 2.5.1

Divide each term in $5 y = \arcsin (- x - 8)$ by $5$ .

$\frac{5 y}{5} = \frac{\arcsin (- x - 8)}{5}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $5$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{\arcsin (- x - 8)}{5}$

Step 2.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arcsin (- x - 8)}{5}$

Step 3

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

$f^{- 1} (x) = \frac{\arcsin (- x - 8)}{5}$

Step 4

Verify if $f^{- 1} (x) = \frac{\arcsin (- x - 8)}{5}$ is the inverse of $f (x) = - \sin (5 x) - 8$ .

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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 $f^{- 1} (- \sin (5 x) - 8)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \sin (5 x) - 8) = \frac{\arcsin (- (- \sin (5 x) - 8) - 8)}{5}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Apply the distributive property.

$f^{- 1} (- \sin (5 x) - 8) = \frac{\arcsin (\sin (5 x) + 8 - 8)}{5}$

Step 4.2.3.1.2

Multiply $- - \sin (5 x)$ .

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Step 4.2.3.1.2.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \sin (5 x) - 8) = \frac{\arcsin (1 \sin (5 x) + 8 - 8)}{5}$

Step 4.2.3.1.2.2

Multiply $\sin (5 x)$ by $1$ .

$f^{- 1} (- \sin (5 x) - 8) = \frac{\arcsin (\sin (5 x) + 8 - 8)}{5}$

Step 4.2.3.1.3

Multiply $- 1$ by $- 8$ .

$f^{- 1} (- \sin (5 x) - 8) = \frac{\arcsin (\sin (5 x) + 8 - 8)}{5}$

Step 4.2.3.2

Combine the opposite terms in $\sin (5 x) + 8 - 8$ .

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Step 4.2.3.2.1

Subtract $8$ from $8$ .

$f^{- 1} (- \sin (5 x) - 8) = \frac{\arcsin (\sin (5 x) + 0)}{5}$

Step 4.2.3.2.2

Add $\sin (5 x)$ and $0$ .

$f^{- 1} (- \sin (5 x) - 8) = \frac{\arcsin (\sin (5 x))}{5}$

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 $f (\frac{\arcsin (- x - 8)}{5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arcsin (- x - 8)}{5}) = - \sin (5 (\frac{\arcsin (- x - 8)}{5})) - 8$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Cancel the common factor of $5$ .

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Step 4.3.3.1.1

Cancel the common factor.

$f (\frac{\arcsin (- x - 8)}{5}) = - \sin (5 (\frac{\arcsin (- x - 8)}{5})) - 8$

Step 4.3.3.1.2

Rewrite the expression.

$f (\frac{\arcsin (- x - 8)}{5}) = - \sin (\arcsin (- x - 8)) - 8$

Step 4.3.3.2

The functions sine and arcsine are inverses.

$f (\frac{\arcsin (- x - 8)}{5}) = - (- x - 8) - 8$

Step 4.3.3.3

Apply the distributive property.

$f (\frac{\arcsin (- x - 8)}{5}) = x + 8 - 8$

Step 4.3.3.4

Multiply $- - x$ .

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Step 4.3.3.4.1

Multiply $- 1$ by $- 1$ .

$f (\frac{\arcsin (- x - 8)}{5}) = 1 x + 8 - 8$

Step 4.3.3.4.2

Multiply $x$ by $1$ .

$f (\frac{\arcsin (- x - 8)}{5}) = x + 8 - 8$

Step 4.3.3.5

Multiply $- 1$ by $- 8$ .

$f (\frac{\arcsin (- x - 8)}{5}) = x + 8 - 8$

Step 4.3.4

Combine the opposite terms in $x + 8 - 8$ .

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Step 4.3.4.1

Subtract $8$ from $8$ .

$f (\frac{\arcsin (- x - 8)}{5}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\frac{\arcsin (- x - 8)}{5}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arcsin (- x - 8)}{5}$ is the inverse of $f (x) = - \sin (5 x) - 8$ .

$f^{- 1} (x) = \frac{\arcsin (- x - 8)}{5}$