Precalculus Examples

cos (5 x) = 0

$\cos (5 x) = 0$

Step 1

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

$x$ from inside the cosine.

$5 x = \arccos (0)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

$\arccos (0)$ is

$\frac{π}{2}$ .

$5 x = \frac{π}{2}$

Step 3

Divide each term in

$5 x = \frac{π}{2}$ by

$5$ and simplify.

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Step 3.1

Divide each term in

$5 x = \frac{π}{2}$ by

$5$ .

$\frac{5 x}{5} = \frac{\frac{π}{2}}{5}$

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.

$\frac{5 x}{5} = \frac{\frac{π}{2}}{5}$

Step 3.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{π}{2}}{5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{5}$

Step 3.3.2

Multiply

$\frac{π}{2} \cdot \frac{1}{5}$ .

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Step 3.3.2.1

Multiply

$\frac{π}{2}$ by

$\frac{1}{5}$ .

$x = \frac{π}{2 \cdot 5}$

Step 3.3.2.2

Multiply

$2$ by

$5$ .

$x = \frac{π}{10}$

Step 4

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from

$2 π$ to find the solution in the fourth quadrant.

$5 x = 2 π - \frac{π}{2}$

Step 5

Solve for

$x$ .

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Step 5.1

Simplify.

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Step 5.1.1

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$5 x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 5.1.2

Combine

$2 π$ and

$\frac{2}{2}$ .

$5 x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 5.1.3

Combine the numerators over the common denominator.

$5 x = \frac{2 π \cdot 2 - π}{2}$

Step 5.1.4

Multiply

$2$ by

$2$ .

$5 x = \frac{4 π - π}{2}$

Step 5.1.5

Subtract

$π$ from

$4 π$ .

$5 x = \frac{3 π}{2}$

Step 5.2

Divide each term in

$5 x = \frac{3 π}{2}$ by

$5$ and simplify.

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Step 5.2.1

Divide each term in

$5 x = \frac{3 π}{2}$ by

$5$ .

$\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5}$

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.

$\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5}$

Step 5.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{3 π}{2}}{5}$

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.

$x = \frac{3 π}{2} \cdot \frac{1}{5}$

Step 5.2.3.2

Multiply

$\frac{3 π}{2} \cdot \frac{1}{5}$ .

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Step 5.2.3.2.1

Multiply

$\frac{3 π}{2}$ by

$\frac{1}{5}$ .

$x = \frac{3 π}{2 \cdot 5}$

Step 5.2.3.2.2

Multiply

$2$ by

$5$ .

$x = \frac{3 π}{10}$

Step 6

Find the period of

$\cos (5 x)$ .

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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

$\cos (5 x)$ function is

$\frac{2 π}{5}$ so values will repeat every

$\frac{2 π}{5}$ radians in both directions.

$x = \frac{π}{10} + \frac{2 π n}{5}, \frac{3 π}{10} + \frac{2 π n}{5}$ , for any integer

$n$

Step 8

Consolidate the answers.

$x = \frac{π}{10} + \frac{π n}{5}$ , for any integer

$n$