Calculus Examples

$0 = \frac{π}{2} + π \cos (\frac{π x}{2})$

Step 1

Rewrite the equation as $\frac{π}{2} + π \cos (\frac{π x}{2}) = 0$ .

$\frac{π}{2} + π \cos (\frac{π x}{2}) = 0$

Step 2

Subtract $\frac{π}{2}$ from both sides of the equation.

$π \cos (\frac{π x}{2}) = - \frac{π}{2}$

Step 3

Divide each term in $π \cos (\frac{π x}{2}) = - \frac{π}{2}$ by $π$ and simplify.

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Step 3.1

Divide each term in $π \cos (\frac{π x}{2}) = - \frac{π}{2}$ by $π$ .

$\frac{π \cos (\frac{π x}{2})}{π} = \frac{- \frac{π}{2}}{π}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $π$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{π \cos (\frac{π x}{2})}{π} = \frac{- \frac{π}{2}}{π}$

Step 3.2.1.2

Divide $\cos (\frac{π x}{2})$ by $1$ .

$\cos (\frac{π x}{2}) = \frac{- \frac{π}{2}}{π}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (\frac{π x}{2}) = - \frac{π}{2} \cdot \frac{1}{π}$

Step 3.3.2

Cancel the common factor of $π$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$\cos (\frac{π x}{2}) = \frac{- π}{2} \cdot \frac{1}{π}$

Step 3.3.2.2

Factor $π$ out of $- π$ .

$\cos (\frac{π x}{2}) = \frac{π \cdot - 1}{2} \cdot \frac{1}{π}$

Step 3.3.2.3

Cancel the common factor.

$\cos (\frac{π x}{2}) = \frac{π \cdot - 1}{2} \cdot \frac{1}{π}$

Step 3.3.2.4

Rewrite the expression.

$\cos (\frac{π x}{2}) = \frac{- 1}{2}$

Step 3.3.3

Move the negative in front of the fraction.

$\cos (\frac{π x}{2}) = - \frac{1}{2}$

Step 4

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

$\frac{π x}{2} = \arccos (- \frac{1}{2})$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$\frac{π x}{2} = \frac{2 π}{3}$

Step 6

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

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

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Simplify $\frac{2}{π} \cdot \frac{π x}{2}$ .

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Step 7.1.1.1

Cancel the common factor of $2$ .

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Step 7.1.1.1.1

Cancel the common factor.

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

Step 7.1.1.1.2

Rewrite the expression.

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

Step 7.1.1.2

Cancel the common factor of $π$ .

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Step 7.1.1.2.1

Factor $π$ out of $π x$ .

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

Step 7.1.1.2.2

Cancel the common factor.

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

Step 7.1.1.2.3

Rewrite the expression.

$x = \frac{2}{π} \cdot \frac{2 π}{3}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{2}{π} \cdot \frac{2 π}{3}$ .

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Step 7.2.1.1

Cancel the common factor of $π$ .

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Step 7.2.1.1.1

Factor $π$ out of $2 π$ .

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

Step 7.2.1.1.2

Cancel the common factor.

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

Step 7.2.1.1.3

Rewrite the expression.

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

Step 7.2.1.2

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

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

Step 7.2.1.3

Multiply $2$ by $2$ .

$x = \frac{4}{3}$

Step 8

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$\frac{π x}{2} = 2 π - \frac{2 π}{3}$

Step 9

Solve for $x$ .

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Step 9.1

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

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

Step 9.2

Simplify both sides of the equation.

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Step 9.2.1

Simplify the left side.

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Step 9.2.1.1

Simplify $\frac{2}{π} \cdot \frac{π x}{2}$ .

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Step 9.2.1.1.1

Cancel the common factor of $2$ .

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Step 9.2.1.1.1.1

Cancel the common factor.

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

Step 9.2.1.1.1.2

Rewrite the expression.

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

Step 9.2.1.1.2

Cancel the common factor of $π$ .

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Step 9.2.1.1.2.1

Factor $π$ out of $π x$ .

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

Step 9.2.1.1.2.2

Cancel the common factor.

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

Step 9.2.1.1.2.3

Rewrite the expression.

$x = \frac{2}{π} (2 π - \frac{2 π}{3})$

Step 9.2.2

Simplify the right side.

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Step 9.2.2.1

Simplify $\frac{2}{π} (2 π - \frac{2 π}{3})$ .

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Step 9.2.2.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 9.2.2.1.2

Combine fractions.

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Step 9.2.2.1.2.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 9.2.2.1.2.2

Combine the numerators over the common denominator.

$x = \frac{2}{π} \cdot \frac{2 π \cdot 3 - 2 π}{3}$

Step 9.2.2.1.3

Simplify the numerator.

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Step 9.2.2.1.3.1

Multiply $3$ by $2$ .

$x = \frac{2}{π} \cdot \frac{6 π - 2 π}{3}$

Step 9.2.2.1.3.2

Subtract $2 π$ from $6 π$ .

$x = \frac{2}{π} \cdot \frac{4 π}{3}$

Step 9.2.2.1.4

Simplify terms.

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Step 9.2.2.1.4.1

Cancel the common factor of $π$ .

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Step 9.2.2.1.4.1.1

Factor $π$ out of $4 π$ .

$x = \frac{2}{π} \cdot \frac{π \cdot 4}{3}$

Step 9.2.2.1.4.1.2

Cancel the common factor.

$x = \frac{2}{π} \cdot \frac{π \cdot 4}{3}$

Step 9.2.2.1.4.1.3

Rewrite the expression.

$x = 2 (\frac{4}{3})$

Step 9.2.2.1.4.2

Combine $2$ and $\frac{4}{3}$ .

$x = \frac{2 \cdot 4}{3}$

Step 9.2.2.1.4.3

Multiply $2$ by $4$ .

$x = \frac{8}{3}$

Step 10

Find the period of $\cos (\frac{π x}{2})$ .

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Step 10.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 10.2

Replace $b$ with $\frac{π}{2}$ in the formula for period.

$\frac{2 π}{| \frac{π}{2} |}$

Step 10.3

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 10.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 10.5

Cancel the common factor of $π$ .

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Step 10.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 10.5.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 10.5.3

Rewrite the expression.

$2 \cdot 2$

Step 10.6

Multiply $2$ by $2$ .

$4$

Step 11

The period of the $\cos (\frac{π x}{2})$ function is $4$ so values will repeat every $4$ radians in both directions.

$x = \frac{4}{3} + 4 n, \frac{8}{3} + 4 n$ , for any integer $n$