Trigonometry Examples

$y = \frac{1}{3} \cdot \sin (\frac{1}{2} x)$

Step 1

Interchange the variables.

$x = \frac{1}{3} \cdot \sin (\frac{1}{2} y)$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{1}{3} \cdot \sin (\frac{1}{2} y) = x$ .

$\frac{1}{3} \cdot \sin (\frac{1}{2} y) = x$

Step 2.2

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} \sin (\frac{1}{2} y)) = 3 x$

Step 2.3

Simplify the left side.

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Step 2.3.1

Simplify $3 (\frac{1}{3} \sin (\frac{1}{2} y))$ .

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Step 2.3.1.1

Combine $\frac{1}{3}$ and $\sin (\frac{1}{2} y)$ .

$3 \frac{\sin (\frac{1}{2} y)}{3} = 3 x$

Step 2.3.1.2

Cancel the common factor of $3$ .

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Step 2.3.1.2.1

Cancel the common factor.

$3 \frac{\sin (\frac{1}{2} y)}{3} = 3 x$

Step 2.3.1.2.2

Rewrite the expression.

$\sin (\frac{1}{2} y) = 3 x$

Step 2.4

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

$\frac{1}{2} y = \arcsin (3 x)$

Step 2.5

Simplify the left side.

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Step 2.5.1

Combine $\frac{1}{2}$ and $y$ .

$\frac{y}{2} = \arcsin (3 x)$

Step 2.6

Multiply both sides of the equation by $2$ .

$2 \frac{y}{2} = 2 \arcsin (3 x)$

Step 2.7

Simplify the left side.

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Step 2.7.1

Cancel the common factor of $2$ .

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Step 2.7.1.1

Cancel the common factor.

$2 \frac{y}{2} = 2 \arcsin (3 x)$

Step 2.7.1.2

Rewrite the expression.

$y = 2 \arcsin (3 x)$

Step 3

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

$f^{- 1} (x) = 2 \arcsin (3 x)$

Step 4

Verify if $f^{- 1} (x) = 2 \arcsin (3 x)$ is the inverse of $f (x) = \frac{1}{3} \cdot \sin (\frac{1}{2} x)$ .

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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 $f^{- 1} (\frac{1}{3} \cdot \sin (\frac{1}{2} x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{3} \cdot \sin (\frac{1}{2} x)) = 2 \arcsin (3 (\frac{1}{3} \cdot \sin (\frac{1}{2} x)))$

Step 4.2.3

Combine $\frac{1}{2}$ and $x$ .

$f^{- 1} (\frac{1}{3} \cdot \sin (\frac{1}{2} x)) = 2 \arcsin (3 (\frac{1}{3} \cdot \sin (\frac{x}{2})))$

Step 4.2.4

Combine $\frac{1}{3}$ and $\sin (\frac{x}{2})$ .

$f^{- 1} (\frac{1}{3} \cdot \sin (\frac{1}{2} x)) = 2 \arcsin (3 (\frac{\sin (\frac{x}{2})}{3}))$

Step 4.2.5

Cancel the common factor of $3$ .

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Step 4.2.5.1

Cancel the common factor.

$f^{- 1} (\frac{1}{3} \cdot \sin (\frac{1}{2} x)) = 2 \arcsin (3 (\frac{\sin (\frac{x}{2})}{3}))$

Step 4.2.5.2

Rewrite the expression.

$f^{- 1} (\frac{1}{3} \cdot \sin (\frac{1}{2} x)) = 2 \arcsin (\sin (\frac{x}{2}))$

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 $f (2 \arcsin (3 x))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (2 \arcsin (3 x)) = \frac{1}{3} \cdot \sin (\frac{1}{2} \cdot (2 \arcsin (3 x)))$

Step 4.3.3

Cancel the common factor of $2$ .

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Step 4.3.3.1

Factor $2$ out of $2 \arcsin (3 x)$ .

$f (2 \arcsin (3 x)) = \frac{1}{3} \cdot \sin (\frac{1}{2} \cdot (2 (\arcsin (3 x))))$

Step 4.3.3.2

Cancel the common factor.

$f (2 \arcsin (3 x)) = \frac{1}{3} \cdot \sin (\frac{1}{2} \cdot (2 \arcsin (3 x)))$

Step 4.3.3.3

Rewrite the expression.

$f (2 \arcsin (3 x)) = \frac{1}{3} \cdot \sin (\arcsin (3 x))$

Step 4.3.4

The functions sine and arcsine are inverses.

$f (2 \arcsin (3 x)) = \frac{1}{3} \cdot (3 x)$

Step 4.3.5

Cancel the common factor of $3$ .

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Step 4.3.5.1

Factor $3$ out of $3 x$ .

$f (2 \arcsin (3 x)) = \frac{1}{3} \cdot (3 (x))$

Step 4.3.5.2

Cancel the common factor.

$f (2 \arcsin (3 x)) = \frac{1}{3} \cdot (3 x)$

Step 4.3.5.3

Rewrite the expression.

$f (2 \arcsin (3 x)) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 2 \arcsin (3 x)$ is the inverse of $f (x) = \frac{1}{3} \cdot \sin (\frac{1}{2} x)$ .

$f^{- 1} (x) = 2 \arcsin (3 x)$