Find the Roots (Zeros) 2cos(x-2pi)

$2 \cos (x - 2 π)$

Set $2 \cos (x - 2 π)$ equal to $0$ .

$2 \cos (x - 2 π) = 0$

Solve for $x$ .

Divide each term in $2 \cos (x - 2 π) = 0$ by $2$ and simplify.

Divide each term in $2 \cos (x - 2 π) = 0$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $\cos (x - 2 π)$ by $1$ .

$\cos (x - 2 π) = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$\cos (x - 2 π) = 0$

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

$x - 2 π = \arccos (0)$

Simplify the right side.

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

$x - 2 π = \frac{π}{2}$

Move all terms not containing $x$ to the right side of the equation.

Add $2 π$ to both sides of the equation.

$x = \frac{π}{2} + 2 π$

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

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

Combine $2 π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$x = \frac{π + 2 π \cdot 2}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$x = \frac{π + 4 π}{2}$

Add $π$ and $4 π$ .

$x = \frac{5 π}{2}$

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.

$x - 2 π = 2 π - \frac{π}{2}$

Solve for $x$ .

Simplify $2 π - \frac{π}{2}$ .

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

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

Combine fractions.

Combine $2 π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$x - 2 π = \frac{2 π \cdot 2 - π}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$x - 2 π = \frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$x - 2 π = \frac{3 π}{2}$

Move all terms not containing $x$ to the right side of the equation.

Add $2 π$ to both sides of the equation.

$x = \frac{3 π}{2} + 2 π$

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

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

Combine $2 π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$x = \frac{3 π + 2 π \cdot 2}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$x = \frac{3 π + 4 π}{2}$

Add $3 π$ and $4 π$ .

$x = \frac{7 π}{2}$

Find the period of $\cos (x - 2 π)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

The period of the $\cos (x - 2 π)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{5 π}{2} + 2 π n, \frac{7 π}{2} + 2 π n$ , for any integer $n$

Consolidate the answers.

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

Find the Roots (Zeros) 2cos(x-2pi)

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