Calculus Examples

$y = 2 π - 5 - π \cos (π t)$

Step 1

Rewrite the equation as $2 π - 5 - π \cos (π t) = y$ .

$2 π - 5 - π \cos (π t) = y$

Step 2

Move all terms not containing $\cos (π t)$ to the right side of the equation.

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Step 2.1

Subtract $2 π$ from both sides of the equation.

$- 5 - π \cos (π t) = y - 2 π$

Step 2.2

Add $5$ to both sides of the equation.

$- π \cos (π t) = y - 2 π + 5$

Step 3

Divide each term in $- π \cos (π t) = y - 2 π + 5$ by $- π$ and simplify.

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Step 3.1

Divide each term in $- π \cos (π t) = y - 2 π + 5$ by $- π$ .

$\frac{- π \cos (π t)}{- π} = \frac{y}{- π} + \frac{- 2 π}{- π} + \frac{5}{- π}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{π \cos (π t)}{π} = \frac{y}{- π} + \frac{- 2 π}{- π} + \frac{5}{- π}$

Step 3.2.2

Cancel the common factor of $π$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{π \cos (π t)}{π} = \frac{y}{- π} + \frac{- 2 π}{- π} + \frac{5}{- π}$

Step 3.2.2.2

Divide $\cos (π t)$ by $1$ .

$\cos (π t) = \frac{y}{- π} + \frac{- 2 π}{- π} + \frac{5}{- π}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative in front of the fraction.

$\cos (π t) = - \frac{y}{π} + \frac{- 2 π}{- π} + \frac{5}{- π}$

Step 3.3.1.2

Cancel the common factor of $π$ .

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Step 3.3.1.2.1

Cancel the common factor.

$\cos (π t) = - \frac{y}{π} + \frac{- 2 π}{- π} + \frac{5}{- π}$

Step 3.3.1.2.2

Rewrite the expression.

$\cos (π t) = - \frac{y}{π} + \frac{- 2}{- 1} + \frac{5}{- π}$

Step 3.3.1.2.3

Move the negative one from the denominator of $\frac{- 2}{- 1}$ .

$\cos (π t) = - \frac{y}{π} - 1 \cdot - 2 + \frac{5}{- π}$

Step 3.3.1.3

Rewrite $- 1 \cdot - 2$ as $- - 2$ .

$\cos (π t) = - \frac{y}{π} - - 2 + \frac{5}{- π}$

Step 3.3.1.4

Multiply $- 1$ by $- 2$ .

$\cos (π t) = - \frac{y}{π} + 2 + \frac{5}{- π}$

Step 3.3.1.5

Move the negative in front of the fraction.

$\cos (π t) = - \frac{y}{π} + 2 - \frac{5}{π}$

Step 4

Take the inverse cosine of both sides of the equation to extract $t$ from inside the cosine.

$π t = \arccos (- \frac{y}{π} + 2 - \frac{5}{π})$

Step 5

Divide each term in $π t = \arccos (- \frac{y}{π} + 2 - \frac{5}{π})$ by $π$ and simplify.

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Step 5.1

Divide each term in $π t = \arccos (- \frac{y}{π} + 2 - \frac{5}{π})$ by $π$ .

$\frac{π t}{π} = \frac{\arccos (- \frac{y}{π} + 2 - \frac{5}{π})}{π}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $π$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{π t}{π} = \frac{\arccos (- \frac{y}{π} + 2 - \frac{5}{π})}{π}$

Step 5.2.1.2

Divide $t$ by $1$ .

$t = \frac{\arccos (- \frac{y}{π} + 2 - \frac{5}{π})}{π}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify the numerator.

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Step 5.3.1.1

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

$t = \frac{\arccos (- \frac{y}{π} + \frac{2 π}{π} - \frac{5}{π})}{π}$

Step 5.3.1.2

Combine the numerators over the common denominator.

$t = \frac{\arccos (- \frac{y}{π} + \frac{2 π - 5}{π})}{π}$