Calculus Examples

$\frac{- 5 π \sin (\frac{π x}{8})}{8} = 0$

Step 1

Set the numerator equal to zero.

$- 5 π \sin (\frac{π x}{8}) = 0$

Step 2

Solve the equation for $x$ .

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Step 2.1

Divide each term in $- 5 π \sin (\frac{π x}{8}) = 0$ by $- 5 π$ and simplify.

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Step 2.1.1

Divide each term in $- 5 π \sin (\frac{π x}{8}) = 0$ by $- 5 π$ .

$\frac{- 5 π \sin (\frac{π x}{8})}{- 5 π} = \frac{0}{- 5 π}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $- 5$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{- 5 π \sin (\frac{π x}{8})}{- 5 π} = \frac{0}{- 5 π}$

Step 2.1.2.1.2

Rewrite the expression.

$\frac{π \sin (\frac{π x}{8})}{π} = \frac{0}{- 5 π}$

Step 2.1.2.2

Cancel the common factor of $π$ .

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Step 2.1.2.2.1

Cancel the common factor.

$\frac{π \sin (\frac{π x}{8})}{π} = \frac{0}{- 5 π}$

Step 2.1.2.2.2

Divide $\sin (\frac{π x}{8})$ by $1$ .

$\sin (\frac{π x}{8}) = \frac{0}{- 5 π}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Cancel the common factor of $0$ and $- 5$ .

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Step 2.1.3.1.1

Factor $5$ out of $0$ .

$\sin (\frac{π x}{8}) = \frac{5 (0)}{- 5 π}$

Step 2.1.3.1.2

Cancel the common factors.

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Step 2.1.3.1.2.1

Factor $5$ out of $- 5 π$ .

$\sin (\frac{π x}{8}) = \frac{5 (0)}{5 (- π)}$

Step 2.1.3.1.2.2

Cancel the common factor.

$\sin (\frac{π x}{8}) = \frac{5 \cdot 0}{5 (- π)}$

Step 2.1.3.1.2.3

Rewrite the expression.

$\sin (\frac{π x}{8}) = \frac{0}{- π}$

Step 2.1.3.2

Divide $0$ by $- π$ .

$\sin (\frac{π x}{8}) = 0$

Step 2.2

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

$\frac{π x}{8} = \arcsin (0)$

Step 2.3

Simplify the right side.

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Step 2.3.1

The exact value of $\arcsin (0)$ is $0$ .

$\frac{π x}{8} = 0$

Step 2.4

Set the numerator equal to zero.

$π x = 0$

Step 2.5

Divide each term in $π x = 0$ by $π$ and simplify.

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Step 2.5.1

Divide each term in $π x = 0$ by $π$ .

$\frac{π x}{π} = \frac{0}{π}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $π$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{π}$

Step 2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{π}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Divide $0$ by $π$ .

$x = 0$

Step 2.6

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$\frac{π x}{8} = π - 0$

Step 2.7

Solve for $x$ .

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Step 2.7.1

Multiply both sides of the equation by $\frac{8}{π}$ .

$\frac{8}{π} \cdot \frac{π x}{8} = \frac{8}{π} (π - 0)$

Step 2.7.2

Simplify both sides of the equation.

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Step 2.7.2.1

Simplify the left side.

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Step 2.7.2.1.1

Simplify $\frac{8}{π} \cdot \frac{π x}{8}$ .

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Step 2.7.2.1.1.1

Cancel the common factor of $8$ .

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Step 2.7.2.1.1.1.1

Cancel the common factor.

$\frac{8}{π} \cdot \frac{π x}{8} = \frac{8}{π} (π - 0)$

Step 2.7.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{8}{π} (π - 0)$

Step 2.7.2.1.1.2

Cancel the common factor of $π$ .

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Step 2.7.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{8}{π} (π - 0)$

Step 2.7.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{8}{π} (π - 0)$

Step 2.7.2.1.1.2.3

Rewrite the expression.

$x = \frac{8}{π} (π - 0)$

Step 2.7.2.2

Simplify the right side.

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Step 2.7.2.2.1

Simplify $\frac{8}{π} (π - 0)$ .

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Step 2.7.2.2.1.1

Subtract $0$ from $π$ .

$x = \frac{8}{π} π$

Step 2.7.2.2.1.2

Cancel the common factor of $π$ .

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Step 2.7.2.2.1.2.1

Cancel the common factor.

$x = \frac{8}{π} π$

Step 2.7.2.2.1.2.2

Rewrite the expression.

$x = 8$

Step 2.8

Find the period of $\sin (\frac{π x}{8})$ .

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Step 2.8.1

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

$\frac{2 π}{| b |}$

Step 2.8.2

Replace $b$ with $\frac{π}{8}$ in the formula for period.

$\frac{2 π}{| \frac{π}{8} |}$

Step 2.8.3

$\frac{π}{8}$ is approximately $0.39269908$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{8}}$

Step 2.8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{8}{π}$

Step 2.8.5

Cancel the common factor of $π$ .

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Step 2.8.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{8}{π}$

Step 2.8.5.2

Cancel the common factor.

$π \cdot 2 \frac{8}{π}$

Step 2.8.5.3

Rewrite the expression.

$2 \cdot 8$

Step 2.8.6

Multiply $2$ by $8$ .

$16$

Step 2.9

The period of the $\sin (\frac{π x}{8})$ function is $16$ so values will repeat every $16$ radians in both directions.

$x = 16 n, 8 + 16 n$ , for any integer $n$

Step 3

Consolidate the answers.

$x = 8 n$ , for any integer $n$