Finite Math Examples

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

Step 1

Set $(\frac{π}{3}) (- \sin (\frac{π x}{3}))$ equal to $0$ .

$(\frac{π}{3}) (- \sin (\frac{π x}{3})) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $\frac{1}{\frac{π}{3} \cdot - 1}$ .

$\frac{1}{\frac{π}{3} \cdot - 1} (\frac{π}{3} \cdot (- 1 \sin (\frac{π x}{3}))) = \frac{1}{\frac{π}{3} \cdot - 1} \cdot 0$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $\frac{1}{\frac{π}{3} \cdot - 1} (\frac{π}{3} \cdot (- 1 \sin (\frac{π x}{3})))$ .

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Step 2.2.1.1.1

Cancel the common factor of $\frac{π}{3}$ .

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Step 2.2.1.1.1.1

Factor $\frac{π}{3}$ out of $\frac{π}{3} \cdot - 1$ .

$\frac{1}{\frac{π}{3} (- 1)} (\frac{π}{3} \cdot (- 1 \sin (\frac{π x}{3}))) = \frac{1}{\frac{π}{3} \cdot - 1} \cdot 0$

Step 2.2.1.1.1.2

Cancel the common factor.

$\frac{1}{\frac{π}{3} \cdot - 1} (\frac{π}{3} (- 1 \sin (\frac{π x}{3}))) = \frac{1}{\frac{π}{3} \cdot - 1} \cdot 0$

Step 2.2.1.1.1.3

Rewrite the expression.

$\frac{1}{- 1} (- 1 \sin (\frac{π x}{3})) = \frac{1}{\frac{π}{3} \cdot - 1} \cdot 0$

Step 2.2.1.1.2

Combine $\frac{1}{- 1}$ and $\sin (\frac{π x}{3})$ .

$- 1 \frac{\sin (\frac{π x}{3})}{- 1} = \frac{1}{\frac{π}{3} \cdot - 1} \cdot 0$

Step 2.2.1.1.3

Simplify the expression.

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Step 2.2.1.1.3.1

Move the negative one from the denominator of $\frac{\sin (\frac{π x}{3})}{- 1}$ .

$- 1 (- 1 \cdot \sin (\frac{π x}{3})) = \frac{1}{\frac{π}{3} \cdot - 1} \cdot 0$

Step 2.2.1.1.3.2

Rewrite $- 1 \cdot \sin (\frac{π x}{3})$ as $- \sin (\frac{π x}{3})$ .

$- 1 (- \sin (\frac{π x}{3})) = \frac{1}{\frac{π}{3} \cdot - 1} \cdot 0$

Step 2.2.1.1.4

Multiply $- 1 (- \sin (\frac{π x}{3}))$ .

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Step 2.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$1 \sin (\frac{π x}{3}) = \frac{1}{\frac{π}{3} \cdot - 1} \cdot 0$

Step 2.2.1.1.4.2

Multiply $\sin (\frac{π x}{3})$ by $1$ .

$\sin (\frac{π x}{3}) = \frac{1}{\frac{π}{3} \cdot - 1} \cdot 0$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $\frac{1}{\frac{π}{3} \cdot - 1} \cdot 0$ .

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Step 2.2.2.1.1

Combine $\frac{π}{3}$ and $- 1$ .

$\sin (\frac{π x}{3}) = \frac{1}{\frac{π \cdot - 1}{3}} \cdot 0$

Step 2.2.2.1.2

Simplify the denominator.

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Step 2.2.2.1.2.1

Move $- 1$ to the left of $π$ .

$\sin (\frac{π x}{3}) = \frac{1}{\frac{- 1 \cdot π}{3}} \cdot 0$

Step 2.2.2.1.2.2

Move the negative in front of the fraction.

$\sin (\frac{π x}{3}) = \frac{1}{- \frac{π}{3}} \cdot 0$

Step 2.2.2.1.3

Cancel the common factor of $1$ and $- 1$ .

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Step 2.2.2.1.3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\sin (\frac{π x}{3}) = \frac{- 1 (- 1)}{- \frac{π}{3}} \cdot 0$

Step 2.2.2.1.3.2

Move the negative in front of the fraction.

$\sin (\frac{π x}{3}) = - \frac{1}{\frac{π}{3}} \cdot 0$

Step 2.2.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$\sin (\frac{π x}{3}) = - (1 \frac{3}{π}) \cdot 0$

Step 2.2.2.1.5

Multiply $\frac{3}{π}$ by $1$ .

$\sin (\frac{π x}{3}) = - \frac{3}{π} \cdot 0$

Step 2.2.2.1.6

Multiply $- \frac{3}{π} \cdot 0$ .

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Step 2.2.2.1.6.1

Multiply $0$ by $- 1$ .

$\sin (\frac{π x}{3}) = 0 \frac{3}{π}$

Step 2.2.2.1.6.2

Multiply $0$ by $\frac{3}{π}$ .

$\sin (\frac{π x}{3}) = 0$

Step 2.3

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

$\frac{π x}{3} = \arcsin (0)$

Step 2.4

Simplify the right side.

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Step 2.4.1

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

$\frac{π x}{3} = 0$

Step 2.5

Set the numerator equal to zero.

$π x = 0$

Step 2.6

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

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Step 2.6.1

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

$\frac{π x}{π} = \frac{0}{π}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $π$ .

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Step 2.6.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{π}$

Step 2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{π}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Divide $0$ by $π$ .

$x = 0$

Step 2.7

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.

$\frac{π x}{3} = π - 0$

Step 2.8

Solve for $x$ .

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Step 2.8.1

Multiply both sides of the equation by $\frac{3}{π}$ .

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} (π - 0)$

Step 2.8.2

Simplify both sides of the equation.

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Step 2.8.2.1

Simplify the left side.

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Step 2.8.2.1.1

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

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Step 2.8.2.1.1.1

Cancel the common factor of $3$ .

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Step 2.8.2.1.1.1.1

Cancel the common factor.

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} (π - 0)$

Step 2.8.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{3}{π} (π - 0)$

Step 2.8.2.1.1.2

Cancel the common factor of $π$ .

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Step 2.8.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{3}{π} (π - 0)$

Step 2.8.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{3}{π} (π - 0)$

Step 2.8.2.1.1.2.3

Rewrite the expression.

$x = \frac{3}{π} (π - 0)$

Step 2.8.2.2

Simplify the right side.

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Step 2.8.2.2.1

Simplify $\frac{3}{π} (π - 0)$ .

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Step 2.8.2.2.1.1

Subtract $0$ from $π$ .

$x = \frac{3}{π} π$

Step 2.8.2.2.1.2

Cancel the common factor of $π$ .

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Step 2.8.2.2.1.2.1

Cancel the common factor.

$x = \frac{3}{π} π$

Step 2.8.2.2.1.2.2

Rewrite the expression.

$x = 3$

Step 2.9

Find the period of $\sin (\frac{π x}{3})$ .

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Step 2.9.1

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

$\frac{2 π}{| b |}$

Step 2.9.2

Replace $b$ with $\frac{π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{π}{3} |}$

Step 2.9.3

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Step 2.9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Step 2.9.5

Cancel the common factor of $π$ .

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Step 2.9.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Step 2.9.5.2

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Step 2.9.5.3

Rewrite the expression.

$2 \cdot 3$

Step 2.9.6

Multiply $2$ by $3$ .

$6$

Step 2.10

The period of the $\sin (\frac{π x}{3})$ function is $6$ so values will repeat every $6$ radians in both directions.

$x = 6 n, 3 + 6 n$ , for any integer $n$

Step 2.11

Consolidate the answers.

$x = 3 n$ , for any integer $n$

Step 3