Precalculus Examples

sin (9 x) = 1

$\sin (9 x) = 1$

Step 1

Take the inverse sine of both sides of the equation to extract

x

$x$ from inside the sine.

9 x = arcsin (1)

$9 x = \arcsin (1)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

arcsin (1)

$\arcsin (1)$ is

\frac{π}{2}

$\frac{π}{2}$ .

9 x = \frac{π}{2}

$9 x = \frac{π}{2}$

9 x = \frac{π}{2}

$9 x = \frac{π}{2}$

Step 3

Divide each term in

9 x = \frac{π}{2}

$9 x = \frac{π}{2}$ by

9

$9$ and simplify.

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Step 3.1

Divide each term in

9 x = \frac{π}{2}

$9 x = \frac{π}{2}$ by

9

$9$ .

\frac{9 x}{9} = \frac{\frac{π}{2}}{9}

$\frac{9 x}{9} = \frac{\frac{π}{2}}{9}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

9

$9$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{\frac{π}{2}}{9}$

Step 3.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{π}{2}}{9}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{9}$

Step 3.3.2

Multiply

$\frac{π}{2} \cdot \frac{1}{9}$ .

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Step 3.3.2.1

Multiply

$\frac{π}{2}$ by

$\frac{1}{9}$ .

$x = \frac{π}{2 \cdot 9}$

Step 3.3.2.2

Multiply

$2$ by

$9$ .

$x = \frac{π}{18}$

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.

$9 x = π - \frac{π}{2}$

Step 5

Solve for

$x$ .

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Step 5.1

Simplify.

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Step 5.1.1

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$9 x = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 5.1.2

Combine

$π$ and

$\frac{2}{2}$ .

$9 x = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 5.1.3

Combine the numerators over the common denominator.

$9 x = \frac{π \cdot 2 - π}{2}$

Step 5.1.4

Subtract

$π$ from

$π \cdot 2$ .

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Step 5.1.4.1

Reorder

$π$ and

$2$ .

$9 x = \frac{2 \cdot π - π}{2}$

Step 5.1.4.2

Subtract

$π$ from

$2 \cdot π$ .

$9 x = \frac{π}{2}$

Step 5.2

Divide each term in

$9 x = \frac{π}{2}$ by

$9$ and simplify.

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Step 5.2.1

Divide each term in

$9 x = \frac{π}{2}$ by

$9$ .

$\frac{9 x}{9} = \frac{\frac{π}{2}}{9}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

$9$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{\frac{π}{2}}{9}$

Step 5.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{π}{2}}{9}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{9}$

Step 5.2.3.2

Multiply

$\frac{π}{2} \cdot \frac{1}{9}$ .

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Step 5.2.3.2.1

Multiply

$\frac{π}{2}$ by

$\frac{1}{9}$ .

$x = \frac{π}{2 \cdot 9}$

Step 5.2.3.2.2

Multiply

$2$ by

$9$ .

$x = \frac{π}{18}$

Step 6

Find the period of

$\sin (9 x)$ .

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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

$9$ in the formula for period.

$\frac{2 π}{| 9 |}$

Step 6.3

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

$0$ and

$9$ is

$9$ .

$\frac{2 π}{9}$

Step 7

The period of the

$\sin (9 x)$ function is

$\frac{2 π}{9}$ so values will repeat every

$\frac{2 π}{9}$ radians in both directions.

$x = \frac{π}{18} + \frac{2 π n}{9}$ , for any integer

$n$