Precalculus Examples

$\sin (5 x) = 0$

Step 1

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

$5 x = \arcsin (0)$

Step 2

Simplify the right side.

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Step 2.1

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

$5 x = 0$

Step 3

Divide each term in $5 x = 0$ by $5$ and simplify.

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Step 3.1

Divide each term in $5 x = 0$ by $5$ .

$\frac{5 x}{5} = \frac{0}{5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{0}{5}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $5$ .

$x = 0$

Step 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.

$5 x = π - 0$

Step 5

Solve for $x$ .

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Step 5.1

Simplify.

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Step 5.1.1

Multiply $- 1$ by $0$ .

$5 x = π + 0$

Step 5.1.2

Add $π$ and $0$ .

$5 x = π$

Step 5.2

Divide each term in $5 x = π$ by $5$ and simplify.

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Step 5.2.1

Divide each term in $5 x = π$ by $5$ .

$\frac{5 x}{5} = \frac{π}{5}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{π}{5}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{5}$

Step 6

Find the period of $\sin (5 x)$ .

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

Replace $b$ with $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

Step 6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

Step 7

The period of the $\sin (5 x)$ function is $\frac{2 π}{5}$ so values will repeat every $\frac{2 π}{5}$ radians in both directions.

$x = \frac{2 π n}{5}, \frac{π}{5} + \frac{2 π n}{5}$ , for any integer $n$

Step 8

Consolidate the answers.

$x = \frac{π n}{5}$ , for any integer $n$