Trigonometry Examples

$6 y = 5 x$

Step 1

Divide each term in $6 y = 5 x$ by $6$ and simplify.

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Step 1.1

Divide each term in $6 y = 5 x$ by $6$ .

$\frac{6 y}{6} = \frac{5 x}{6}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $6$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{5 x}{6}$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 x}{6}$

Step 2

Interchange the variables.

$x = \frac{5 y}{6}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{5 y}{6} = x$ .

$\frac{5 y}{6} = x$

Step 3.2

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} \cdot \frac{5 y}{6} = \frac{6}{5} x$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{6}{5} \cdot \frac{5 y}{6}$ .

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Step 3.3.1.1.1

Cancel the common factor of $6$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 y}{6} = \frac{6}{5} x$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 y) = \frac{6}{5} x$

Step 3.3.1.1.2

Cancel the common factor of $5$ .

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Step 3.3.1.1.2.1

Factor $5$ out of $5 y$ .

$\frac{1}{5} (5 (y)) = \frac{6}{5} x$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 y) = \frac{6}{5} x$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{6}{5} x$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Combine $\frac{6}{5}$ and $x$ .

$y = \frac{6 x}{5}$

Step 4

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

$f^{- 1} (x) = \frac{6 x}{5}$

Step 5

Verify if $f^{- 1} (x) = \frac{6 x}{5}$ is the inverse of $f (x) = \frac{5 x}{6}$ .

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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 $f^{- 1} (\frac{5 x}{6})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5 x}{6}) = \frac{6 (\frac{5 x}{6})}{5}$

Step 5.2.3

Combine $6$ and $\frac{5 x}{6}$ .

$f^{- 1} (\frac{5 x}{6}) = \frac{\frac{6 (5 x)}{6}}{5}$

Step 5.2.4

Multiply $6$ by $5$ .

$f^{- 1} (\frac{5 x}{6}) = \frac{\frac{30 x}{6}}{5}$

Step 5.2.5

Reduce the expression by cancelling the common factors.

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Step 5.2.5.1

Reduce the expression $\frac{30 x}{6}$ by cancelling the common factors.

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Step 5.2.5.1.1

Factor $6$ out of $30 x$ .

$f^{- 1} (\frac{5 x}{6}) = \frac{\frac{6 (5 x)}{6}}{5}$

Step 5.2.5.1.2

Factor $6$ out of $6$ .

$f^{- 1} (\frac{5 x}{6}) = \frac{\frac{6 (5 x)}{6 (1)}}{5}$

Step 5.2.5.1.3

Cancel the common factor.

$f^{- 1} (\frac{5 x}{6}) = \frac{\frac{6 (5 x)}{6 \cdot 1}}{5}$

Step 5.2.5.1.4

Rewrite the expression.

$f^{- 1} (\frac{5 x}{6}) = \frac{\frac{5 x}{1}}{5}$

Step 5.2.5.2

Divide $5 x$ by $1$ .

$f^{- 1} (\frac{5 x}{6}) = \frac{5 x}{5}$

Step 5.2.6

Cancel the common factor of $5$ .

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Step 5.2.6.1

Cancel the common factor.

$f^{- 1} (\frac{5 x}{6}) = \frac{5 x}{5}$

Step 5.2.6.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{5 x}{6}) = x$

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 $f (\frac{6 x}{5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{6 x}{5}) = \frac{5 (\frac{6 x}{5})}{6}$

Step 5.3.3

Combine $5$ and $\frac{6 x}{5}$ .

$f (\frac{6 x}{5}) = \frac{\frac{5 (6 x)}{5}}{6}$

Step 5.3.4

Multiply $5$ by $6$ .

$f (\frac{6 x}{5}) = \frac{\frac{30 x}{5}}{6}$

Step 5.3.5

Reduce the expression by cancelling the common factors.

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Step 5.3.5.1

Reduce the expression $\frac{30 x}{5}$ by cancelling the common factors.

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Step 5.3.5.1.1

Factor $5$ out of $30 x$ .

$f (\frac{6 x}{5}) = \frac{\frac{5 (6 x)}{5}}{6}$

Step 5.3.5.1.2

Factor $5$ out of $5$ .

$f (\frac{6 x}{5}) = \frac{\frac{5 (6 x)}{5 (1)}}{6}$

Step 5.3.5.1.3

Cancel the common factor.

$f (\frac{6 x}{5}) = \frac{\frac{5 (6 x)}{5 \cdot 1}}{6}$

Step 5.3.5.1.4

Rewrite the expression.

$f (\frac{6 x}{5}) = \frac{\frac{6 x}{1}}{6}$

Step 5.3.5.2

Divide $6 x$ by $1$ .

$f (\frac{6 x}{5}) = \frac{6 x}{6}$

Step 5.3.6

Cancel the common factor of $6$ .

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Step 5.3.6.1

Cancel the common factor.

$f (\frac{6 x}{5}) = \frac{6 x}{6}$

Step 5.3.6.2

Divide $x$ by $1$ .

$f (\frac{6 x}{5}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{6 x}{5}$ is the inverse of $f (x) = \frac{5 x}{6}$ .

$f^{- 1} (x) = \frac{6 x}{5}$