Precalculus Examples

$f (x) = \frac{\sin (π x)}{x}$

Step 1

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

$\frac{\sin (π x)}{x} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Set the numerator equal to zero.

$\sin (π x) = 0$

Step 2.2

Solve the equation for $x$ .

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Step 2.2.1

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

$π x = \arcsin (0)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

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

$π x = 0$

Step 2.2.3

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

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Step 2.2.3.1

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

$\frac{π x}{π} = \frac{0}{π}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $π$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{π}$

Step 2.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{π}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $0$ by $π$ .

$x = 0$

Step 2.2.4

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.

$π x = π - 0$

Step 2.2.5

Solve for $x$ .

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Step 2.2.5.1

Simplify.

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Step 2.2.5.1.1

Multiply $- 1$ by $0$ .

$π x = π + 0$

Step 2.2.5.1.2

Add $π$ and $0$ .

$π x = π$

Step 2.2.5.2

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

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Step 2.2.5.2.1

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

$\frac{π x}{π} = \frac{π}{π}$

Step 2.2.5.2.2

Simplify the left side.

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Step 2.2.5.2.2.1

Cancel the common factor of $π$ .

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Step 2.2.5.2.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{π}{π}$

Step 2.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{π}$

Step 2.2.5.2.3

Simplify the right side.

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Step 2.2.5.2.3.1

Cancel the common factor of $π$ .

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Step 2.2.5.2.3.1.1

Cancel the common factor.

$x = \frac{π}{π}$

Step 2.2.5.2.3.1.2

Rewrite the expression.

$x = 1$

Step 2.2.6

Find the period of $\sin (π x)$ .

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Step 2.2.6.1

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

$\frac{2 π}{| b |}$

Step 2.2.6.2

Replace $b$ with $π$ in the formula for period.

$\frac{2 π}{| π |}$

Step 2.2.6.3

$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π}$

Step 2.2.6.4

Cancel the common factor of $π$ .

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Step 2.2.6.4.1

Cancel the common factor.

$\frac{2 π}{π}$

Step 2.2.6.4.2

Divide $2$ by $1$ .

$2$

Step 2.2.7

The period of the $\sin (π x)$ function is $2$ so values will repeat every $2$ radians in both directions.

$x = 2 n, 1 + 2 n$ , for any integer $n$

Step 2.3

Consolidate the answers.

$x = 1 n$ , for any integer $n$

Step 3