Calculus Examples

$π \sin (π x) = 0$

Step 1

Divide each term in $π \sin (π x) = 0$ by $π$ and simplify.

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Step 1.1

Divide each term in $π \sin (π x) = 0$ by $π$ .

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $π$ .

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Divide $\sin (π x)$ by $1$ .

$\sin (π x) = \frac{0}{π}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $0$ by $π$ .

$\sin (π x) = 0$

Step 2

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

$π x = \arcsin (0)$

Step 3

Simplify the right side.

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Step 3.1

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

$π x = 0$

Step 4

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

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Step 4.1

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

$\frac{π x}{π} = \frac{0}{π}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $π$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{π}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{π}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $π$ .

$x = 0$

Step 5

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 6

Solve for $x$ .

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Step 6.1

Simplify.

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Step 6.1.1

Multiply $- 1$ by $0$ .

$π x = π + 0$

Step 6.1.2

Add $π$ and $0$ .

$π x = π$

Step 6.2

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

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Step 6.2.1

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

$\frac{π x}{π} = \frac{π}{π}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $π$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{π}{π}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{π}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Cancel the common factor of $π$ .

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Step 6.2.3.1.1

Cancel the common factor.

$x = \frac{π}{π}$

Step 6.2.3.1.2

Rewrite the expression.

$x = 1$

Step 7

Find the period of $\sin (π x)$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| π |}$

Step 7.3

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

$\frac{2 π}{π}$

Step 7.4

Cancel the common factor of $π$ .

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Step 7.4.1

Cancel the common factor.

$\frac{2 π}{π}$

Step 7.4.2

Divide $2$ by $1$ .

$2$

Step 8

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 9

Consolidate the answers.

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