Algebra Examples

$y = - 3 \cos (6 x)$

Step 1

Set $- 3 \cos (6 x)$ equal to $0$ .

$- 3 \cos (6 x) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Divide each term in $- 3 \cos (6 x) = 0$ by $- 3$ and simplify.

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Step 2.1.1

Divide each term in $- 3 \cos (6 x) = 0$ by $- 3$ .

$\frac{- 3 \cos (6 x)}{- 3} = \frac{0}{- 3}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $- 3$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{- 3 \cos (6 x)}{- 3} = \frac{0}{- 3}$

Step 2.1.2.1.2

Divide $\cos (6 x)$ by $1$ .

$\cos (6 x) = \frac{0}{- 3}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $- 3$ .

$\cos (6 x) = 0$

Step 2.2

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

$6 x = \arccos (0)$

Step 2.3

Simplify the right side.

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Step 2.3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$6 x = \frac{π}{2}$

Step 2.4

Divide each term in $6 x = \frac{π}{2}$ by $6$ and simplify.

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Step 2.4.1

Divide each term in $6 x = \frac{π}{2}$ by $6$ .

$\frac{6 x}{6} = \frac{\frac{π}{2}}{6}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $6$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{\frac{π}{2}}{6}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{6}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{6}$

Step 2.4.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{6}$ .

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Step 2.4.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{6}$ .

$x = \frac{π}{2 \cdot 6}$

Step 2.4.3.2.2

Multiply $2$ by $6$ .

$x = \frac{π}{12}$

Step 2.5

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.

$6 x = 2 π - \frac{π}{2}$

Step 2.6

Solve for $x$ .

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Step 2.6.1

Simplify.

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Step 2.6.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 2.6.1.2

Combine $2 π$ and $\frac{2}{2}$ .

$6 x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 2.6.1.3

Combine the numerators over the common denominator.

$6 x = \frac{2 π \cdot 2 - π}{2}$

Step 2.6.1.4

Multiply $2$ by $2$ .

$6 x = \frac{4 π - π}{2}$

Step 2.6.1.5

Subtract $π$ from $4 π$ .

$6 x = \frac{3 π}{2}$

Step 2.6.2

Divide each term in $6 x = \frac{3 π}{2}$ by $6$ and simplify.

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Step 2.6.2.1

Divide each term in $6 x = \frac{3 π}{2}$ by $6$ .

$\frac{6 x}{6} = \frac{\frac{3 π}{2}}{6}$

Step 2.6.2.2

Simplify the left side.

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Step 2.6.2.2.1

Cancel the common factor of $6$ .

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Step 2.6.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{\frac{3 π}{2}}{6}$

Step 2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{6}$

Step 2.6.2.3

Simplify the right side.

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Step 2.6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{6}$

Step 2.6.2.3.2

Cancel the common factor of $3$ .

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Step 2.6.2.3.2.1

Factor $3$ out of $3 π$ .

$x = \frac{3 (π)}{2} \cdot \frac{1}{6}$

Step 2.6.2.3.2.2

Factor $3$ out of $6$ .

$x = \frac{3 (π)}{2} \cdot \frac{1}{3 (2)}$

Step 2.6.2.3.2.3

Cancel the common factor.

$x = \frac{3 π}{2} \cdot \frac{1}{3 \cdot 2}$

Step 2.6.2.3.2.4

Rewrite the expression.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 2.6.2.3.3

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2}$

Step 2.6.2.3.4

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 2.7

Find the period of $\cos (6 x)$ .

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Step 2.7.1

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

$\frac{2 π}{| b |}$

Step 2.7.2

Replace $b$ with $6$ in the formula for period.

$\frac{2 π}{| 6 |}$

Step 2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{2 π}{6}$

Step 2.7.4

Cancel the common factor of $2$ and $6$ .

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Step 2.7.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{6}$

Step 2.7.4.2

Cancel the common factors.

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Step 2.7.4.2.1

Factor $2$ out of $6$ .

$\frac{2 π}{2 \cdot 3}$

Step 2.7.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 3}$

Step 2.7.4.2.3

Rewrite the expression.

$\frac{π}{3}$

Step 2.8

The period of the $\cos (6 x)$ function is $\frac{π}{3}$ so values will repeat every $\frac{π}{3}$ radians in both directions.

$x = \frac{π}{12} + \frac{π n}{3}, \frac{π}{4} + \frac{π n}{3}$ , for any integer $n$

Step 2.9

Consolidate the answers.

$x = \frac{π}{12} + \frac{π n}{6}$ , for any integer $n$

Step 3