Trigonometry Examples

$f (t) = 40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55$

Step 1

Write $f (t) = 40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55$ as an equation.

$y = 40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55$

Step 2

Interchange the variables.

$t = 40 \sin (\frac{5 π}{4} y - \frac{π}{2}) + 55$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $40 \sin (\frac{5 π}{4} y - \frac{π}{2}) + 55 = t$ .

$40 \sin (\frac{5 π}{4} y - \frac{π}{2}) + 55 = t$

Step 3.2

Subtract $55$ from both sides of the equation.

$40 \sin (\frac{5 π}{4} y - \frac{π}{2}) = t - 55$

Step 3.3

Divide each term in $40 \sin (\frac{5 π}{4} y - \frac{π}{2}) = t - 55$ by $40$ and simplify.

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Step 3.3.1

Divide each term in $40 \sin (\frac{5 π}{4} y - \frac{π}{2}) = t - 55$ by $40$ .

$\frac{40 \sin (\frac{5 π}{4} y - \frac{π}{2})}{40} = \frac{t}{40} + \frac{- 55}{40}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $40$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{40 \sin (\frac{5 π}{4} y - \frac{π}{2})}{40} = \frac{t}{40} + \frac{- 55}{40}$

Step 3.3.2.1.2

Divide $\sin (\frac{5 π}{4} y - \frac{π}{2})$ by $1$ .

$\sin (\frac{5 π}{4} y - \frac{π}{2}) = \frac{t}{40} + \frac{- 55}{40}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Cancel the common factor of $- 55$ and $40$ .

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Step 3.3.3.1.1.1

Factor $5$ out of $- 55$ .

$\sin (\frac{5 π}{4} y - \frac{π}{2}) = \frac{t}{40} + \frac{5 (- 11)}{40}$

Step 3.3.3.1.1.2

Cancel the common factors.

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Step 3.3.3.1.1.2.1

Factor $5$ out of $40$ .

$\sin (\frac{5 π}{4} y - \frac{π}{2}) = \frac{t}{40} + \frac{5 \cdot - 11}{5 \cdot 8}$

Step 3.3.3.1.1.2.2

Cancel the common factor.

$\sin (\frac{5 π}{4} y - \frac{π}{2}) = \frac{t}{40} + \frac{5 \cdot - 11}{5 \cdot 8}$

Step 3.3.3.1.1.2.3

Rewrite the expression.

$\sin (\frac{5 π}{4} y - \frac{π}{2}) = \frac{t}{40} + \frac{- 11}{8}$

Step 3.3.3.1.2

Move the negative in front of the fraction.

$\sin (\frac{5 π}{4} y - \frac{π}{2}) = \frac{t}{40} - \frac{11}{8}$

Step 3.4

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

$\frac{5 π}{4} y - \frac{π}{2} = \arcsin (\frac{t}{40} - \frac{11}{8})$

Step 3.5

Simplify the left side.

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Step 3.5.1

Combine $\frac{5 π}{4}$ and $y$ .

$\frac{5 π y}{4} - \frac{π}{2} = \arcsin (\frac{t}{40} - \frac{11}{8})$

Step 3.6

Add $\frac{π}{2}$ to both sides of the equation.

$\frac{5 π y}{4} = \arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2}$

Step 3.7

Multiply both sides of the equation by $\frac{4}{5 π}$ .

$\frac{4}{5 π} \cdot \frac{5 π y}{4} = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.8

Simplify both sides of the equation.

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Step 3.8.1

Simplify the left side.

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Step 3.8.1.1

Simplify $\frac{4}{5 π} \cdot \frac{5 π y}{4}$ .

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Step 3.8.1.1.1

Cancel the common factor of $4$ .

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Step 3.8.1.1.1.1

Cancel the common factor.

$\frac{4}{5 π} \cdot \frac{5 π y}{4} = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.8.1.1.1.2

Rewrite the expression.

$\frac{1}{5 π} (5 π y) = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.8.1.1.2

Cancel the common factor of $5 π$ .

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Step 3.8.1.1.2.1

Factor $5 π$ out of $5 π y$ .

$\frac{1}{5 π} (5 π (y)) = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.8.1.1.2.2

Cancel the common factor.

$\frac{1}{5 π} (5 π y) = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.8.1.1.2.3

Rewrite the expression.

$y = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.8.2

Simplify the right side.

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Step 3.8.2.1

Simplify $\frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$ .

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Step 3.8.2.1.1

Apply the distributive property.

$y = \frac{4}{5 π} \arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{4}{5 π} \cdot \frac{π}{2}$

Step 3.8.2.1.2

Combine $\frac{4}{5 π}$ and $\arcsin (\frac{t}{40} - \frac{11}{8})$ .

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8})}{5 π} + \frac{4}{5 π} \cdot \frac{π}{2}$

Step 3.8.2.1.3

Cancel the common factor of $2$ .

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Step 3.8.2.1.3.1

Factor $2$ out of $4$ .

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8})}{5 π} + \frac{2 (2)}{5 π} \cdot \frac{π}{2}$

Step 3.8.2.1.3.2

Cancel the common factor.

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8})}{5 π} + \frac{2 \cdot 2}{5 π} \cdot \frac{π}{2}$

Step 3.8.2.1.3.3

Rewrite the expression.

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8})}{5 π} + \frac{2}{5 π} π$

Step 3.8.2.1.4

Cancel the common factor of $π$ .

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Step 3.8.2.1.4.1

Factor $π$ out of $5 π$ .

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8})}{5 π} + \frac{2}{π \cdot 5} π$

Step 3.8.2.1.4.2

Cancel the common factor.

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8})}{5 π} + \frac{2}{π \cdot 5} π$

Step 3.8.2.1.4.3

Rewrite the expression.

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8})}{5 π} + \frac{2}{5}$

Step 3.9

Add $\frac{π}{2}$ to both sides of the equation.

$\frac{5 π y}{4} = \arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2}$

Step 3.10

Multiply both sides of the equation by $\frac{4}{5 π}$ .

$\frac{4}{5 π} \cdot \frac{5 π y}{4} = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.11

Simplify both sides of the equation.

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Step 3.11.1

Simplify the left side.

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Step 3.11.1.1

Simplify $\frac{4}{5 π} \cdot \frac{5 π y}{4}$ .

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Step 3.11.1.1.1

Cancel the common factor of $4$ .

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Step 3.11.1.1.1.1

Cancel the common factor.

$\frac{4}{5 π} \cdot \frac{5 π y}{4} = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.11.1.1.1.2

Rewrite the expression.

$\frac{1}{5 π} (5 π y) = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.11.1.1.2

Cancel the common factor of $5 π$ .

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Step 3.11.1.1.2.1

Factor $5 π$ out of $5 π y$ .

$\frac{1}{5 π} (5 π (y)) = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.11.1.1.2.2

Cancel the common factor.

$\frac{1}{5 π} (5 π y) = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.11.1.1.2.3

Rewrite the expression.

$y = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$

Step 3.11.2

Simplify the right side.

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Step 3.11.2.1

Simplify $\frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{π}{2})$ .

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Step 3.11.2.1.1

To write $\arcsin (\frac{t}{40} - \frac{11}{8})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{4}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) \cdot \frac{2}{2} + \frac{π}{2})$

Step 3.11.2.1.2

Simplify terms.

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Step 3.11.2.1.2.1

Combine $\arcsin (\frac{t}{40} - \frac{11}{8})$ and $\frac{2}{2}$ .

$y = \frac{4}{5 π} (\frac{\arcsin (\frac{t}{40} - \frac{11}{8}) \cdot 2}{2} + \frac{π}{2})$

Step 3.11.2.1.2.2

Combine the numerators over the common denominator.

$y = \frac{4}{5 π} \cdot \frac{\arcsin (\frac{t}{40} - \frac{11}{8}) \cdot 2 + π}{2}$

Step 3.11.2.1.2.3

Cancel the common factor of $2$ .

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Step 3.11.2.1.2.3.1

Factor $2$ out of $4$ .

$y = \frac{2 (2)}{5 π} \cdot \frac{\arcsin (\frac{t}{40} - \frac{11}{8}) \cdot 2 + π}{2}$

Step 3.11.2.1.2.3.2

Cancel the common factor.

$y = \frac{2 \cdot 2}{5 π} \cdot \frac{\arcsin (\frac{t}{40} - \frac{11}{8}) \cdot 2 + π}{2}$

Step 3.11.2.1.2.3.3

Rewrite the expression.

$y = \frac{2}{5 π} (\arcsin (\frac{t}{40} - \frac{11}{8}) \cdot 2 + π)$

Step 3.11.2.1.3

Move $2$ to the left of $\arcsin (\frac{t}{40} - \frac{11}{8})$ .

$y = \frac{2}{5 π} (2 \arcsin (\frac{t}{40} - \frac{11}{8}) + π)$

Step 3.11.2.1.4

Apply the distributive property.

$y = \frac{2}{5 π} (2 \arcsin (\frac{t}{40} - \frac{11}{8})) + \frac{2}{5 π} π$

Step 3.11.2.1.5

Multiply $\frac{2}{5 π} (2 \arcsin (\frac{t}{40} - \frac{11}{8}))$ .

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Step 3.11.2.1.5.1

Combine $2$ and $\frac{2}{5 π}$ .

$y = \frac{2 \cdot 2}{5 π} \arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{2}{5 π} π$

Step 3.11.2.1.5.2

Multiply $2$ by $2$ .

$y = \frac{4}{5 π} \arcsin (\frac{t}{40} - \frac{11}{8}) + \frac{2}{5 π} π$

Step 3.11.2.1.5.3

Combine $\frac{4}{5 π}$ and $\arcsin (\frac{t}{40} - \frac{11}{8})$ .

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8})}{5 π} + \frac{2}{5 π} π$

Step 3.11.2.1.6

Cancel the common factor of $π$ .

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Step 3.11.2.1.6.1

Factor $π$ out of $5 π$ .

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8})}{5 π} + \frac{2}{π \cdot 5} π$

Step 3.11.2.1.6.2

Cancel the common factor.

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8})}{5 π} + \frac{2}{π \cdot 5} π$

Step 3.11.2.1.6.3

Rewrite the expression.

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8})}{5 π} + \frac{2}{5}$

Step 3.11.2.1.7

Simplify the numerator.

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Step 3.11.2.1.7.1

To write $- \frac{11}{8}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11}{8} \cdot \frac{5}{5})}{5 π} + \frac{2}{5}$

Step 3.11.2.1.7.2

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.11.2.1.7.2.1

Multiply $\frac{11}{8}$ by $\frac{5}{5}$ .

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11 \cdot 5}{8 \cdot 5})}{5 π} + \frac{2}{5}$

Step 3.11.2.1.7.2.2

Multiply $8$ by $5$ .

$y = \frac{4 \arcsin (\frac{t}{40} - \frac{11 \cdot 5}{40})}{5 π} + \frac{2}{5}$

Step 3.11.2.1.7.3

Combine the numerators over the common denominator.

$y = \frac{4 \arcsin (\frac{t - 11 \cdot 5}{40})}{5 π} + \frac{2}{5}$

Step 3.11.2.1.7.4

Multiply $- 11$ by $5$ .

$y = \frac{4 \arcsin (\frac{t - 55}{40})}{5 π} + \frac{2}{5}$

Step 3.11.2.1.8

To write $\frac{2}{5}$ as a fraction with a common denominator, multiply by $\frac{π}{π}$ .

$y = \frac{4 \arcsin (\frac{t - 55}{40})}{5 π} + \frac{2}{5} \cdot \frac{π}{π}$

Step 3.11.2.1.9

Multiply $\frac{2}{5}$ by $\frac{π}{π}$ .

$y = \frac{4 \arcsin (\frac{t - 55}{40})}{5 π} + \frac{2 π}{5 π}$

Step 3.11.2.1.10

Combine the numerators over the common denominator.

$y = \frac{4 \arcsin (\frac{t - 55}{40}) + 2 π}{5 π}$

Step 3.11.2.1.11

Factor $2$ out of $4 \arcsin (\frac{t - 55}{40}) + 2 π$ .

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Step 3.11.2.1.11.1

Factor $2$ out of $4 \arcsin (\frac{t - 55}{40})$ .

$y = \frac{2 (2 \arcsin (\frac{t - 55}{40})) + 2 π}{5 π}$

Step 3.11.2.1.11.2

Factor $2$ out of $2 π$ .

$y = \frac{2 (2 \arcsin (\frac{t - 55}{40})) + 2 (π)}{5 π}$

Step 3.11.2.1.11.3

Factor $2$ out of $2 (2 \arcsin (\frac{t - 55}{40})) + 2 (π)$ .

$y = \frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}$

Step 4

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

$f^{- 1} (t) = \frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}$

Step 5

Verify if $f^{- 1} (t) = \frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}$ is the inverse of $f (t) = 40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{(40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) - 55}{40}) + π)}{5 π}$

Step 5.2.3

Cancel the common factor of $(40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) - 55$ and $40$ .

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Step 5.2.3.1

Factor $5$ out of $40 \sin (\frac{5 π}{4} t - \frac{π}{2})$ .

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{5 (8 \sin (\frac{5 π}{4} t - \frac{π}{2})) + 55 - 55}{40}) + π)}{5 π}$

Step 5.2.3.2

Factor $5$ out of $55$ .

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{5 (8 \sin (\frac{5 π}{4} t - \frac{π}{2})) + 5 (11) - 55}{40}) + π)}{5 π}$

Step 5.2.3.3

Factor $5$ out of $5 (8 \sin (\frac{5 π}{4} t - \frac{π}{2})) + 5 (11)$ .

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{5 (8 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 11) - 55}{40}) + π)}{5 π}$

Step 5.2.3.4

Factor $5$ out of $- 55$ .

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{5 (8 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 11) + 5 (- 11)}{40}) + π)}{5 π}$

Step 5.2.3.5

Factor $5$ out of $5 (8 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 11) + 5 (- 11)$ .

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{5 (8 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 11 - 11)}{40}) + π)}{5 π}$

Step 5.2.3.6

Cancel the common factors.

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Step 5.2.3.6.1

Factor $5$ out of $40$ .

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{5 (8 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 11 - 11)}{5 (8)}) + π)}{5 π}$

Step 5.2.3.6.2

Cancel the common factor.

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{5 (8 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 11 - 11)}{5 \cdot 8}) + π)}{5 π}$

Step 5.2.3.6.3

Rewrite the expression.

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{8 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 11 - 11}{8}) + π)}{5 π}$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

Combine $\frac{5 π}{4}$ and $t$ .

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{8 \sin (\frac{5 π t}{4} - \frac{π}{2}) + 11 - 11}{8}) + π)}{5 π}$

Step 5.2.4.2

Subtract $11$ from $11$ .

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{8 \sin (\frac{5 π t}{4} - \frac{π}{2}) + 0}{8}) + π)}{5 π}$

Step 5.2.4.3

Add $8 \sin (\frac{5 π t}{4} - \frac{π}{2})$ and $0$ .

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{8 \sin (\frac{5 π t}{4} - \frac{π}{2})}{8}) + π)}{5 π}$

Step 5.2.5

Cancel the common factor of $8$ .

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Step 5.2.5.1

Cancel the common factor.

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\frac{8 \sin (\frac{5 π t}{4} - \frac{π}{2})}{8}) + π)}{5 π}$

Step 5.2.5.2

Divide $\sin (\frac{5 π t}{4} - \frac{π}{2})$ by $1$ .

$f^{- 1} (40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55) = \frac{2 (2 \arcsin (\sin (\frac{5 π t}{4} - \frac{π}{2})) + π)}{5 π}$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{5 π}{4} \cdot (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) - \frac{π}{2}) + 55$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $5 π$ .

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Step 5.3.3.1.1.1

Cancel the common factor.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{5 π}{4} \cdot \frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π} - \frac{π}{2}) + 55$

Step 5.3.3.1.1.2

Rewrite the expression.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{1}{4} \cdot (2 (2 \arcsin (\frac{t - 55}{40}) + π)) - \frac{π}{2}) + 55$

Step 5.3.3.1.2

Cancel the common factor of $2$ .

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Step 5.3.3.1.2.1

Factor $2$ out of $4$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{1}{2 (2)} \cdot (2 (2 \arcsin (\frac{t - 55}{40}) + π)) - \frac{π}{2}) + 55$

Step 5.3.3.1.2.2

Cancel the common factor.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{1}{2 \cdot 2} \cdot (2 (2 \arcsin (\frac{t - 55}{40}) + π)) - \frac{π}{2}) + 55$

Step 5.3.3.1.2.3

Rewrite the expression.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{1}{2} \cdot (2 \arcsin (\frac{t - 55}{40}) + π) - \frac{π}{2}) + 55$

Step 5.3.3.1.3

Apply the distributive property.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{1}{2} \cdot (2 \arcsin (\frac{t - 55}{40})) + \frac{1}{2} \cdot π - \frac{π}{2}) + 55$

Step 5.3.3.1.4

Cancel the common factor of $2$ .

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Step 5.3.3.1.4.1

Factor $2$ out of $2 \arcsin (\frac{t - 55}{40})$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{1}{2} \cdot (2 (\arcsin (\frac{t - 55}{40}))) + \frac{1}{2} \cdot π - \frac{π}{2}) + 55$

Step 5.3.3.1.4.2

Cancel the common factor.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{1}{2} \cdot (2 \arcsin (\frac{t - 55}{40})) + \frac{1}{2} \cdot π - \frac{π}{2}) + 55$

Step 5.3.3.1.4.3

Rewrite the expression.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\arcsin (\frac{t - 55}{40}) + \frac{1}{2} \cdot π - \frac{π}{2}) + 55$

Step 5.3.3.1.5

Combine $\frac{1}{2}$ and $π$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\arcsin (\frac{t - 55}{40}) + \frac{π}{2} - \frac{π}{2}) + 55$

Step 5.3.3.1.6

To write $\arcsin (\frac{t - 55}{40})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\arcsin (\frac{t - 55}{40}) \cdot \frac{2}{2} + \frac{π}{2} - \frac{π}{2}) + 55$

Step 5.3.3.1.7

Combine $\arcsin (\frac{t - 55}{40})$ and $\frac{2}{2}$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{\arcsin (\frac{t - 55}{40}) \cdot 2}{2} + \frac{π}{2} - \frac{π}{2}) + 55$

Step 5.3.3.1.8

Combine the numerators over the common denominator.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{\arcsin (\frac{t - 55}{40}) \cdot 2 + π}{2} - \frac{π}{2}) + 55$

Step 5.3.3.1.9

Move $2$ to the left of $\arcsin (\frac{t - 55}{40})$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{2 \arcsin (\frac{t - 55}{40}) + π}{2} - \frac{π}{2}) + 55$

Step 5.3.3.2

Combine the numerators over the common denominator.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{2 \arcsin (\frac{t - 55}{40}) + π - π}{2}) + 55$

Step 5.3.3.3

Combine the opposite terms in $2 \arcsin (\frac{t - 55}{40}) + π - π$ .

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Step 5.3.3.3.1

Subtract $π$ from $π$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{2 \arcsin (\frac{t - 55}{40}) + 0}{2}) + 55$

Step 5.3.3.3.2

Add $2 \arcsin (\frac{t - 55}{40})$ and $0$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{2 \arcsin (\frac{t - 55}{40})}{2}) + 55$

Step 5.3.3.4

Cancel the common factor of $2$ .

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Step 5.3.3.4.1

Cancel the common factor.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\frac{2 \arcsin (\frac{t - 55}{40})}{2}) + 55$

Step 5.3.3.4.2

Divide $\arcsin (\frac{t - 55}{40})$ by $1$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 \sin (\arcsin (\frac{t - 55}{40})) + 55$

Step 5.3.3.5

The functions sine and arcsine are inverses.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 (\frac{t - 55}{40}) + 55$

Step 5.3.3.6

Cancel the common factor of $40$ .

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Step 5.3.3.6.1

Cancel the common factor.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = 40 (\frac{t - 55}{40}) + 55$

Step 5.3.3.6.2

Rewrite the expression.

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = t - 55 + 55$

Step 5.3.4

Combine the opposite terms in $t - 55 + 55$ .

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Step 5.3.4.1

Add $- 55$ and $55$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = t + 0$

Step 5.3.4.2

Add $t$ and $0$ .

$f (\frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}) = t$

Step 5.4

Since $f^{- 1} (f (t)) = t$ and $f (f^{- 1} (t)) = t$ , then $f^{- 1} (t) = \frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}$ is the inverse of $f (t) = 40 \sin (\frac{5 π}{4} t - \frac{π}{2}) + 55$ .

$f^{- 1} (t) = \frac{2 (2 \arcsin (\frac{t - 55}{40}) + π)}{5 π}$