Find the Roots (Zeros) f(x)=tan(3x)

$f (x) = \tan (3 x)$

Step 1

Set $\tan (3 x)$ equal to $0$ .

$\tan (3 x) = 0$

Step 2

Solve for $x$ .

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

$3 x = \arctan (0)$

Simplify the right side.

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

$3 x = 0$

Divide each term in $3 x = 0$ by $3$ and simplify.

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x = 0$

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.

$3 x = π + 0$

Solve for $x$ .

Add $π$ and $0$ .

$3 x = π$

Divide each term in $3 x = π$ by $3$ and simplify.

Divide each term in $3 x = π$ by $3$ .

$\frac{3 x}{3} = \frac{π}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{π}{3}$

Divide $x$ by $1$ .

$x = \frac{π}{3}$

Find the period of $\tan (3 x)$ .

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

$\frac{π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{π}{3}$

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

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

Consolidate the answers.

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

Step 3

Find the Roots (Zeros) f(x)=tan(3x)

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