Trigonometry Examples

$f (x) = 5 \sin (x) + 3$

Step 1

Write $f (x) = 5 \sin (x) + 3$ as an equation.

$y = 5 \sin (x) + 3$

Step 2

Interchange the variables.

$x = 5 \sin (y) + 3$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $5 \sin (y) + 3 = x$ .

$5 \sin (y) + 3 = x$

Step 3.2

Subtract $3$ from both sides of the equation.

$5 \sin (y) = x - 3$

Step 3.3

Divide each term in $5 \sin (y) = x - 3$ by $5$ and simplify.

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Step 3.3.1

Divide each term in $5 \sin (y) = x - 3$ by $5$ .

$\frac{5 \sin (y)}{5} = \frac{x}{5} + \frac{- 3}{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{5 \sin (y)}{5} = \frac{x}{5} + \frac{- 3}{5}$

Step 3.3.2.1.2

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

$\sin (y) = \frac{x}{5} + \frac{- 3}{5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$\sin (y) = \frac{x}{5} - \frac{3}{5}$

Step 3.4

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

$y = \arcsin (\frac{x}{5} - \frac{3}{5})$

Step 4

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

$f^{- 1} (x) = \arcsin (\frac{x}{5} - \frac{3}{5})$

Step 5

Verify if $f^{- 1} (x) = \arcsin (\frac{x}{5} - \frac{3}{5})$ is the inverse of $f (x) = 5 \sin (x) + 3$ .

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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 $f^{- 1} (5 \sin (x) + 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 \sin (x) + 3) = \arcsin (\frac{5 \sin (x) + 3}{5} - \frac{3}{5})$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (5 \sin (x) + 3) = \arcsin (\frac{5 \sin (x) + 3 - 3}{5})$

Step 5.2.4

Combine the opposite terms in $5 \sin (x) + 3 - 3$ .

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Step 5.2.4.1

Subtract $3$ from $3$ .

$f^{- 1} (5 \sin (x) + 3) = \arcsin (\frac{5 \sin (x) + 0}{5})$

Step 5.2.4.2

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

$f^{- 1} (5 \sin (x) + 3) = \arcsin (\frac{5 \sin (x)}{5})$

Step 5.2.5

Cancel the common factor of $5$ .

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Step 5.2.5.1

Cancel the common factor.

$f^{- 1} (5 \sin (x) + 3) = \arcsin (\frac{5 \sin (x)}{5})$

Step 5.2.5.2

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

$f^{- 1} (5 \sin (x) + 3) = \arcsin (\sin (x))$

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 $f (\arcsin (\frac{x}{5} - \frac{3}{5}))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\arcsin (\frac{x}{5} - \frac{3}{5})) = 5 \sin (\arcsin (\frac{x}{5} - \frac{3}{5})) + 3$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

The functions sine and arcsine are inverses.

$f (\arcsin (\frac{x}{5} - \frac{3}{5})) = 5 (\frac{x}{5} - \frac{3}{5}) + 3$

Step 5.3.3.2

Apply the distributive property.

$f (\arcsin (\frac{x}{5} - \frac{3}{5})) = 5 (\frac{x}{5}) + 5 (- \frac{3}{5}) + 3$

Step 5.3.3.3

Cancel the common factor of $5$ .

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Step 5.3.3.3.1

Cancel the common factor.

$f (\arcsin (\frac{x}{5} - \frac{3}{5})) = 5 (\frac{x}{5}) + 5 (- \frac{3}{5}) + 3$

Step 5.3.3.3.2

Rewrite the expression.

$f (\arcsin (\frac{x}{5} - \frac{3}{5})) = x + 5 (- \frac{3}{5}) + 3$

Step 5.3.3.4

Cancel the common factor of $5$ .

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Step 5.3.3.4.1

Move the leading negative in $- \frac{3}{5}$ into the numerator.

$f (\arcsin (\frac{x}{5} - \frac{3}{5})) = x + 5 (\frac{- 3}{5}) + 3$

Step 5.3.3.4.2

Cancel the common factor.

$f (\arcsin (\frac{x}{5} - \frac{3}{5})) = x + 5 (\frac{- 3}{5}) + 3$

Step 5.3.3.4.3

Rewrite the expression.

$f (\arcsin (\frac{x}{5} - \frac{3}{5})) = x - 3 + 3$

Step 5.3.4

Combine the opposite terms in $x - 3 + 3$ .

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Step 5.3.4.1

Add $- 3$ and $3$ .

$f (\arcsin (\frac{x}{5} - \frac{3}{5})) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\arcsin (\frac{x}{5} - \frac{3}{5})) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \arcsin (\frac{x}{5} - \frac{3}{5})$ is the inverse of $f (x) = 5 \sin (x) + 3$ .

$f^{- 1} (x) = \arcsin (\frac{x}{5} - \frac{3}{5})$