Precalculus Examples

sin (5 x) = 0

$\sin (5 x) = 0$

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

$n$