Trigonometry Examples

$y = \arcsin (\frac{x + 3}{4})$

Step 1

Interchange the variables.

$x = \arcsin (\frac{y + 3}{4})$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\arcsin (\frac{y + 3}{4}) = x$ .

$\arcsin (\frac{y + 3}{4}) = x$

Step 2.2

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

$\frac{y + 3}{4} = \sin (x)$

Step 2.3

Simplify the left side.

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Step 2.3.1

Split the fraction $\frac{y + 3}{4}$ into two fractions.

$\frac{y}{4} + \frac{3}{4} = \sin (x)$

Step 2.4

Subtract $\frac{3}{4}$ from both sides of the equation.

$\frac{y}{4} = \sin (x) - \frac{3}{4}$

Step 2.5

Multiply both sides of the equation by $4$ .

$4 \frac{y}{4} = 4 (\sin (x) - \frac{3}{4})$

Step 2.6

Simplify both sides of the equation.

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Step 2.6.1

Simplify the left side.

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Step 2.6.1.1

Cancel the common factor of $4$ .

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Step 2.6.1.1.1

Cancel the common factor.

$4 \frac{y}{4} = 4 (\sin (x) - \frac{3}{4})$

Step 2.6.1.1.2

Rewrite the expression.

$y = 4 (\sin (x) - \frac{3}{4})$

Step 2.6.2

Simplify the right side.

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Step 2.6.2.1

Simplify $4 (\sin (x) - \frac{3}{4})$ .

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Step 2.6.2.1.1

Apply the distributive property.

$y = 4 \sin (x) + 4 (- \frac{3}{4})$

Step 2.6.2.1.2

Cancel the common factor of $4$ .

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Step 2.6.2.1.2.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$y = 4 \sin (x) + 4 (\frac{- 3}{4})$

Step 2.6.2.1.2.2

Cancel the common factor.

$y = 4 \sin (x) + 4 (\frac{- 3}{4})$

Step 2.6.2.1.2.3

Rewrite the expression.

$y = 4 \sin (x) - 3$

Step 2.7

Subtract $\frac{3}{4}$ from both sides of the equation.

$\frac{y}{4} = \sin (x) - \frac{3}{4}$

Step 2.8

Multiply both sides of the equation by $4$ .

$4 \frac{y}{4} = 4 (\sin (x) - \frac{3}{4})$

Step 2.9

Simplify both sides of the equation.

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Step 2.9.1

Simplify the left side.

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Step 2.9.1.1

Cancel the common factor of $4$ .

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Step 2.9.1.1.1

Cancel the common factor.

$4 \frac{y}{4} = 4 (\sin (x) - \frac{3}{4})$

Step 2.9.1.1.2

Rewrite the expression.

$y = 4 (\sin (x) - \frac{3}{4})$

Step 2.9.2

Simplify the right side.

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Step 2.9.2.1

Simplify $4 (\sin (x) - \frac{3}{4})$ .

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Step 2.9.2.1.1

Apply the distributive property.

$y = 4 \sin (x) + 4 (- \frac{3}{4})$

Step 2.9.2.1.2

Cancel the common factor of $4$ .

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Step 2.9.2.1.2.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$y = 4 \sin (x) + 4 (\frac{- 3}{4})$

Step 2.9.2.1.2.2

Cancel the common factor.

$y = 4 \sin (x) + 4 (\frac{- 3}{4})$

Step 2.9.2.1.2.3

Rewrite the expression.

$y = 4 \sin (x) - 3$

Step 3

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

$f^{- 1} (x) = 4 \sin (x) - 3$

Step 4

Verify if $f^{- 1} (x) = 4 \sin (x) - 3$ is the inverse of $f (x) = \arcsin (\frac{x + 3}{4})$ .

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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 $f^{- 1} (\arcsin (\frac{x + 3}{4}))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\arcsin (\frac{x + 3}{4})) = 4 \sin (\arcsin (\frac{x + 3}{4})) - 3$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

The functions sine and arcsine are inverses.

$f^{- 1} (\arcsin (\frac{x + 3}{4})) = 4 (\frac{x + 3}{4}) - 3$

Step 4.2.3.2

Cancel the common factor of $4$ .

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Step 4.2.3.2.1

Cancel the common factor.

$f^{- 1} (\arcsin (\frac{x + 3}{4})) = 4 (\frac{x + 3}{4}) - 3$

Step 4.2.3.2.2

Rewrite the expression.

$f^{- 1} (\arcsin (\frac{x + 3}{4})) = x + 3 - 3$

Step 4.2.4

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

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Step 4.2.4.1

Subtract $3$ from $3$ .

$f^{- 1} (\arcsin (\frac{x + 3}{4})) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\arcsin (\frac{x + 3}{4})) = x$

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 $f (4 \sin (x) - 3)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (4 \sin (x) - 3) = \arcsin (\frac{(4 \sin (x) - 3) + 3}{4})$

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

Add $- 3$ and $3$ .

$f (4 \sin (x) - 3) = \arcsin (\frac{4 \sin (x) + 0}{4})$

Step 4.3.3.2

Add $4 \sin (x)$ and $0$ .

$f (4 \sin (x) - 3) = \arcsin (\frac{4 \sin (x)}{4})$

Step 4.3.4

Cancel the common factor of $4$ .

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Step 4.3.4.1

Cancel the common factor.

$f (4 \sin (x) - 3) = \arcsin (\frac{4 \sin (x)}{4})$

Step 4.3.4.2

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

$f (4 \sin (x) - 3) = \arcsin (\sin (x))$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 4 \sin (x) - 3$ is the inverse of $f (x) = \arcsin (\frac{x + 3}{4})$ .

$f^{- 1} (x) = 4 \sin (x) - 3$