Trigonometry Examples

$y = \frac{3}{2} x$

Step 1

Interchange the variables.

$x = \frac{3}{2} y$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{3}{2} y = x$ .

$\frac{3}{2} y = x$

Step 2.2

Multiply both sides of the equation by $\frac{2}{3}$ .

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

Step 2.3

Simplify both sides of the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Combine $\frac{3}{2}$ and $y$ .

$\frac{2}{3} \cdot \frac{3 y}{2} = \frac{2}{3} x$

Step 2.3.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.1.1.2.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 y}{2} = \frac{2}{3} x$

Step 2.3.1.1.2.2

Rewrite the expression.

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

Step 2.3.1.1.3

Cancel the common factor of $3$ .

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Step 2.3.1.1.3.1

Factor $3$ out of $3 y$ .

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

Step 2.3.1.1.3.2

Cancel the common factor.

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

Step 2.3.1.1.3.3

Rewrite the expression.

$y = \frac{2}{3} x$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Combine $\frac{2}{3}$ and $x$ .

$y = \frac{2 x}{3}$

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Combine $2$ and $\frac{3}{2}$ .

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

Step 4.2.3.2

Combine $\frac{2 \cdot 3}{2}$ and $x$ .

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

Step 4.2.4

Multiply $2$ by $3$ .

$f^{- 1} (\frac{3}{2} x) = \frac{\frac{6 x}{2}}{3}$

Step 4.2.5

Reduce the expression by cancelling the common factors.

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Step 4.2.5.1

Reduce the expression $\frac{6 x}{2}$ by cancelling the common factors.

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Step 4.2.5.1.1

Factor $2$ out of $6 x$ .

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

Step 4.2.5.1.2

Factor $2$ out of $2$ .

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

Step 4.2.5.1.3

Cancel the common factor.

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

Step 4.2.5.1.4

Rewrite the expression.

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

Step 4.2.5.2

Divide $3 x$ by $1$ .

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

Step 4.2.6

Cancel the common factor of $3$ .

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Step 4.2.6.1

Cancel the common factor.

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

Step 4.2.6.2

Divide $x$ by $1$ .

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

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

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

Step 4.3.3

Cancel the common factor of $3$ .

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Step 4.3.3.1

Cancel the common factor.

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

Step 4.3.3.2

Rewrite the expression.

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

Step 4.3.4

Cancel the common factor of $2$ .

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Step 4.3.4.1

Factor $2$ out of $2 x$ .

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

Step 4.3.4.2

Cancel the common factor.

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

Step 4.3.4.3

Rewrite the expression.

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

Step 4.4

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

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