Find All Complex Solutions tan(2x)=0

$\tan (2 x) = 0$

Step 1

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

$2 x = \arctan (0)$

Step 2

Simplify the right side.

The exact value of $\arctan (0)$ is $0$ .

$2 x = 0$

Step 3

Divide each term in $2 x = 0$ by $2$ and simplify.

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x = 0$

Step 4

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$2 x = π + 0$

Step 5

Solve for $x$ .

Add $π$ and $0$ .

$2 x = π$

Divide each term in $2 x = π$ by $2$ and simplify.

Divide each term in $2 x = π$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Divide $x$ by $1$ .

$x = \frac{π}{2}$

Step 6

Find the period of $\tan (2 x)$ .

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

$\frac{π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

Step 7

The period of the $\tan (2 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

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

Step 8

Consolidate the answers.

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

Find All Complex Solutions tan(2x)=0

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