Trigonometry Examples

$\frac{4}{3} p \cdot 27$

Step 1

Interchange the variables.

$p = \frac{4}{3} y \cdot 27$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{4}{3} y \cdot 27 = p$ .

$\frac{4}{3} y \cdot 27 = p$

Step 2.2

Simplify $\frac{4}{3} y \cdot 27$ .

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Step 2.2.1

Combine $\frac{4}{3}$ and $y$ .

$\frac{4 y}{3} \cdot 27 = p$

Step 2.2.2

Cancel the common factor of $3$ .

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Step 2.2.2.1

Factor $3$ out of $27$ .

$\frac{4 y}{3} \cdot (3 (9)) = p$

Step 2.2.2.2

Cancel the common factor.

$\frac{4 y}{3} \cdot (3 \cdot 9) = p$

Step 2.2.2.3

Rewrite the expression.

$4 y \cdot 9 = p$

Step 2.2.3

Multiply $9$ by $4$ .

$36 y = p$

Step 2.3

Divide each term in $36 y = p$ by $36$ and simplify.

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Step 2.3.1

Divide each term in $36 y = p$ by $36$ .

$\frac{36 y}{36} = \frac{p}{36}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $36$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{36 y}{36} = \frac{p}{36}$

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{p}{36}$

Step 3

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

$f^{- 1} (p) = \frac{p}{36}$

Step 4

Verify if $f^{- 1} (p) = \frac{p}{36}$ is the inverse of $f (p) = \frac{4}{3} p \cdot 27$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (p)) = p$ and $f (f^{- 1} (p)) = p$ .

Step 4.2

Evaluate $f^{- 1} (f (p))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (p))$

Step 4.2.2

Evaluate $f^{- 1} (\frac{4}{3} p \cdot 27)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{\frac{4}{3} p \cdot 27}{36}$

Step 4.2.3

Cancel the common factor of $27$ and $36$ .

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Step 4.2.3.1

Factor $9$ out of $\frac{4}{3} p \cdot 27$ .

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{9 (\frac{4}{3} p \cdot 3)}{36}$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $9$ out of $36$ .

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{9 (\frac{4}{3} p \cdot 3)}{9 (4)}$

Step 4.2.3.2.2

Cancel the common factor.

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{9 (\frac{4}{3} p \cdot 3)}{9 \cdot 4}$

Step 4.2.3.2.3

Rewrite the expression.

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{\frac{4}{3} p \cdot 3}{4}$

Step 4.2.4

Simplify the numerator.

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Step 4.2.4.1

Combine $\frac{4}{3}$ and $p$ .

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{\frac{4 p}{3} \cdot 3}{4}$

Step 4.2.4.2

Combine $\frac{4 p}{3}$ and $3$ .

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{\frac{4 p \cdot 3}{3}}{4}$

Step 4.2.5

Multiply $3$ by $4$ .

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{\frac{12 p}{3}}{4}$

Step 4.2.6

Reduce the expression by cancelling the common factors.

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Step 4.2.6.1

Reduce the expression $\frac{12 p}{3}$ by cancelling the common factors.

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Step 4.2.6.1.1

Factor $3$ out of $12 p$ .

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{\frac{3 (4 p)}{3}}{4}$

Step 4.2.6.1.2

Factor $3$ out of $3$ .

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{\frac{3 (4 p)}{3 (1)}}{4}$

Step 4.2.6.1.3

Cancel the common factor.

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{\frac{3 (4 p)}{3 \cdot 1}}{4}$

Step 4.2.6.1.4

Rewrite the expression.

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{\frac{4 p}{1}}{4}$

Step 4.2.6.2

Divide $4 p$ by $1$ .

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{4 p}{4}$

Step 4.2.7

Cancel the common factor of $4$ .

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Step 4.2.7.1

Cancel the common factor.

$f^{- 1} (\frac{4}{3} p \cdot 27) = \frac{4 p}{4}$

Step 4.2.7.2

Divide $p$ by $1$ .

$f^{- 1} (\frac{4}{3} p \cdot 27) = p$

Step 4.3

Evaluate $f (f^{- 1} (p))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (p))$

Step 4.3.2

Evaluate $f (\frac{p}{36})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{p}{36}) = \frac{4}{3} \cdot (\frac{p}{36}) \cdot 27$

Step 4.3.3

Cancel the common factor of $4$ .

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Step 4.3.3.1

Factor $4$ out of $36$ .

$f (\frac{p}{36}) = \frac{4}{3} \cdot \frac{p}{4 (9)} \cdot 27$

Step 4.3.3.2

Cancel the common factor.

$f (\frac{p}{36}) = \frac{4}{3} \cdot \frac{p}{4 \cdot 9} \cdot 27$

Step 4.3.3.3

Rewrite the expression.

$f (\frac{p}{36}) = \frac{1}{3} \cdot \frac{p}{9} \cdot 27$

Step 4.3.4

Multiply $\frac{1}{3}$ by $\frac{p}{9}$ .

$f (\frac{p}{36}) = \frac{p}{3 \cdot 9} \cdot 27$

Step 4.3.5

Multiply $3$ by $9$ .

$f (\frac{p}{36}) = \frac{p}{27} \cdot 27$

Step 4.3.6

Cancel the common factor of $27$ .

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Step 4.3.6.1

Cancel the common factor.

$f (\frac{p}{36}) = \frac{p}{27} \cdot 27$

Step 4.3.6.2

Rewrite the expression.

$f (\frac{p}{36}) = p$

Step 4.4

Since $f^{- 1} (f (p)) = p$ and $f (f^{- 1} (p)) = p$ , then $f^{- 1} (p) = \frac{p}{36}$ is the inverse of $f (p) = \frac{4}{3} p \cdot 27$ .

$f^{- 1} (p) = \frac{p}{36}$