Find the Roots (Zeros) g(t)=2tan(4pit)

$g (t) = 2 \tan (4 π t)$

Step 1

Set $2 \tan (4 π t)$ equal to $0$ .

$2 \tan (4 π t) = 0$

Step 2

Solve for $x$ .

Divide each term in $2 \tan (4 π t) = 0$ by $2$ and simplify.

Divide each term in $2 \tan (4 π t) = 0$ by $2$ .

$\frac{2 \tan (4 π t)}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \tan (4 π t)}{2} = \frac{0}{2}$

Divide $\tan (4 π t)$ by $1$ .

$\tan (4 π t) = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$\tan (4 π t) = 0$

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

$4 π t = \arctan (0)$

Simplify the right side.

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

$4 π t = 0$

Divide each term in $4 π t = 0$ by $4 π$ and simplify.

Divide each term in $4 π t = 0$ by $4 π$ .

$\frac{4 π t}{4 π} = \frac{0}{4 π}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 π t}{4 π} = \frac{0}{4 π}$

Rewrite the expression.

$\frac{π t}{π} = \frac{0}{4 π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{π t}{π} = \frac{0}{4 π}$

Divide $t$ by $1$ .

$t = \frac{0}{4 π}$

Simplify the right side.

Cancel the common factor of $0$ and $4$ .

Factor $4$ out of $0$ .

$t = \frac{4 (0)}{4 π}$

Cancel the common factors.

Factor $4$ out of $4 π$ .

$t = \frac{4 (0)}{4 (π)}$

Cancel the common factor.

$t = \frac{4 \cdot 0}{4 π}$

Rewrite the expression.

$t = \frac{0}{π}$

Divide $0$ by $π$ .

$t = 0$

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.

$4 π t = π + 0$

Solve for $t$ .

Add $π$ and $0$ .

$4 π t = π$

Divide each term in $4 π t = π$ by $4 π$ and simplify.

Divide each term in $4 π t = π$ by $4 π$ .

$\frac{4 π t}{4 π} = \frac{π}{4 π}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 π t}{4 π} = \frac{π}{4 π}$

Rewrite the expression.

$\frac{π t}{π} = \frac{π}{4 π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{π t}{π} = \frac{π}{4 π}$

Divide $t$ by $1$ .

$t = \frac{π}{4 π}$

Simplify the right side.

Cancel the common factor of $π$ .

Cancel the common factor.

$t = \frac{π}{4 π}$

Rewrite the expression.

$t = \frac{1}{4}$

Find the period of $\tan (4 π t)$ .

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

$\frac{π}{| b |}$

Replace $b$ with $4 π$ in the formula for period.

$\frac{π}{| 4 π |}$

$4 π$ is approximately $12.56637061$ which is positive so remove the absolute value

$\frac{π}{4 π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{π}{4 π}$

Rewrite the expression.

$\frac{1}{4}$

The period of the $\tan (4 π t)$ function is $\frac{1}{4}$ so values will repeat every $\frac{1}{4}$ radians in both directions.

$t = \frac{1 n}{4}, \frac{1}{4} + \frac{1 n}{4}$ , for any integer $n$

Consolidate the answers.

$t = \frac{1 n}{4}$ , for any integer $n$

Step 3

Find the Roots (Zeros) g(t)=2tan(4pit)

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