Find the Roots (Zeros) -3cos(6x)

$- 3 \cos (6 x)$

Step 1

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

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

Step 2

Solve for $x$ .

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

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

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

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Divide $0$ by $- 3$ .

$\cos (6 x) = 0$

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

$6 x = \arccos (0)$

Simplify the right side.

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

$6 x = \frac{π}{2}$

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

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

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{6}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{6}$

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

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

$x = \frac{π}{2 \cdot 6}$

Multiply $2$ by $6$ .

$x = \frac{π}{12}$

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.

$6 x = 2 π - \frac{π}{2}$

Solve for $x$ .

Simplify.

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

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

Combine $2 π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$6 x = \frac{2 π \cdot 2 - π}{2}$

Multiply $2$ by $2$ .

$6 x = \frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$6 x = \frac{3 π}{2}$

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

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

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{6}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{6}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 π$ .

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

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

$x = \frac{π}{2} \cdot \frac{1}{2}$

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

$x = \frac{π}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 6 |}$

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

$\frac{2 π}{6}$

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

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\frac{2 π}{2 \cdot 3}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 3}$

Rewrite the expression.

$\frac{π}{3}$

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 $n$

Consolidate the answers.

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

Step 3

Find the Roots (Zeros) -3cos(6x)

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