Find the Roots (Zeros) -2cos(5/4x)

$- 2 \cos (\frac{5}{4} x)$

Step 1

Set $- 2 \cos (\frac{5}{4} x)$ equal to $0$ .

$- 2 \cos (\frac{5}{4} x) = 0$

Step 2

Solve for $x$ .

Divide each term in $- 2 \cos (\frac{5}{4} x) = 0$ by $- 2$ and simplify.

Divide each term in $- 2 \cos (\frac{5}{4} x) = 0$ by $- 2$ .

$\frac{- 2 \cos (\frac{5}{4} x)}{- 2} = \frac{0}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 \cos (\frac{5}{4} x)}{- 2} = \frac{0}{- 2}$

Divide $\cos (\frac{5}{4} x)$ by $1$ .

$\cos (\frac{5}{4} x) = \frac{0}{- 2}$

Simplify the right side.

Divide $0$ by $- 2$ .

$\cos (\frac{5}{4} x) = 0$

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

$\frac{5}{4} x = \arccos (0)$

Simplify the left side.

Combine $\frac{5}{4}$ and $x$ .

$\frac{5 x}{4} = \arccos (0)$

Simplify the right side.

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

$\frac{5 x}{4} = \frac{π}{2}$

Multiply both sides of the equation by $\frac{4}{5}$ .

$\frac{4}{5} \cdot \frac{5 x}{4} = \frac{4}{5} \cdot \frac{π}{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{4}{5} \cdot \frac{5 x}{4}$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4}{5} \cdot \frac{5 x}{4} = \frac{4}{5} \cdot \frac{π}{2}$

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{4}{5} \cdot \frac{π}{2}$

Cancel the common factor of $5$ .

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{4}{5} \cdot \frac{π}{2}$

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{4}{5} \cdot \frac{π}{2}$

Rewrite the expression.

$x = \frac{4}{5} \cdot \frac{π}{2}$

Simplify the right side.

Simplify $\frac{4}{5} \cdot \frac{π}{2}$ .

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{5} \cdot \frac{π}{2}$

Cancel the common factor.

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

Rewrite the expression.

$x = \frac{2}{5} π$

Combine $\frac{2}{5}$ and $π$ .

$x = \frac{2 π}{5}$

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.

$\frac{5 x}{4} = 2 π - \frac{π}{2}$

Solve for $x$ .

Multiply both sides of the equation by $\frac{4}{5}$ .

$\frac{4}{5} \cdot \frac{5 x}{4} = \frac{4}{5} (2 π - \frac{π}{2})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{4}{5} \cdot \frac{5 x}{4}$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4}{5} \cdot \frac{5 x}{4} = \frac{4}{5} (2 π - \frac{π}{2})$

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{4}{5} (2 π - \frac{π}{2})$

Cancel the common factor of $5$ .

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{4}{5} (2 π - \frac{π}{2})$

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{4}{5} (2 π - \frac{π}{2})$

Rewrite the expression.

$x = \frac{4}{5} (2 π - \frac{π}{2})$

Simplify the right side.

Simplify $\frac{4}{5} (2 π - \frac{π}{2})$ .

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

$x = \frac{4}{5} (2 π \cdot \frac{2}{2} - \frac{π}{2})$

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{4}{5} (\frac{2 π \cdot 2}{2} - \frac{π}{2})$

Combine the numerators over the common denominator.

$x = \frac{4}{5} \cdot \frac{2 π \cdot 2 - π}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$x = \frac{2}{5} (2 π \cdot 2 - π)$

Multiply $2$ by $2$ .

$x = \frac{2}{5} (4 π - π)$

Subtract $π$ from $4 π$ .

$x = \frac{2}{5} (3 π)$

Multiply $\frac{2}{5} (3 π)$ .

Combine $3$ and $\frac{2}{5}$ .

$x = \frac{3 \cdot 2}{5} π$

Multiply $3$ by $2$ .

$x = \frac{6}{5} π$

Combine $\frac{6}{5}$ and $π$ .

$x = \frac{6 π}{5}$

Find the period of $\cos (\frac{5}{4} x)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{5}{4}$ in the formula for period.

$\frac{2 π}{| \frac{5}{4} |}$

$\frac{5}{4}$ is approximately $1.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{5}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{5}$

Multiply $2 π \frac{4}{5}$ .

Combine $\frac{4}{5}$ and $2$ .

$\frac{4 \cdot 2}{5} π$

Multiply $4$ by $2$ .

$\frac{8}{5} π$

Combine $\frac{8}{5}$ and $π$ .

$\frac{8 π}{5}$

The period of the $\cos (\frac{5}{4} x)$ function is $\frac{8 π}{5}$ so values will repeat every $\frac{8 π}{5}$ radians in both directions.

$x = \frac{2 π}{5} + \frac{8 π n}{5}, \frac{6 π}{5} + \frac{8 π n}{5}$ , for any integer $n$

Consolidate the answers.

$x = \frac{2 π}{5} + \frac{4 π n}{5}$ , for any integer $n$

Step 3

Find the Roots (Zeros) -2cos(5/4x)

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