Algebra Examples

$g (t) = 2 \tan (4 π t)$

Step 1

Set $2 \tan (4 π t)$ equal to $0$ .

$2 \tan (4 π t) = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.1.1

Cancel the common factor.

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

Step 2.1.2.1.2

Divide $\tan (4 π t)$ by $1$ .

$\tan (4 π t) = \frac{0}{2}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $2$ .

$\tan (4 π t) = 0$

Step 2.2

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

$4 π t = \arctan (0)$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$4 π t = 0$

Step 2.4

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

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Step 2.4.1

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

$\frac{4 π t}{4 π} = \frac{0}{4 π}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{4 π t}{4 π} = \frac{0}{4 π}$

Step 2.4.2.1.2

Rewrite the expression.

$\frac{π t}{π} = \frac{0}{4 π}$

Step 2.4.2.2

Cancel the common factor of $π$ .

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Step 2.4.2.2.1

Cancel the common factor.

$\frac{π t}{π} = \frac{0}{4 π}$

Step 2.4.2.2.2

Divide $t$ by $1$ .

$t = \frac{0}{4 π}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

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

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Step 2.4.3.1.1

Factor $4$ out of $0$ .

$t = \frac{4 (0)}{4 π}$

Step 2.4.3.1.2

Cancel the common factors.

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Step 2.4.3.1.2.1

Factor $4$ out of $4 π$ .

$t = \frac{4 (0)}{4 (π)}$

Step 2.4.3.1.2.2

Cancel the common factor.

$t = \frac{4 \cdot 0}{4 π}$

Step 2.4.3.1.2.3

Rewrite the expression.

$t = \frac{0}{π}$

Step 2.4.3.2

Divide $0$ by $π$ .

$t = 0$

Step 2.5

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.

$4 π t = π + 0$

Step 2.6

Solve for $t$ .

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Step 2.6.1

Add $π$ and $0$ .

$4 π t = π$

Step 2.6.2

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

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Step 2.6.2.1

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

$\frac{4 π t}{4 π} = \frac{π}{4 π}$

Step 2.6.2.2

Simplify the left side.

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Step 2.6.2.2.1

Cancel the common factor of $4$ .

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Step 2.6.2.2.1.1

Cancel the common factor.

$\frac{4 π t}{4 π} = \frac{π}{4 π}$

Step 2.6.2.2.1.2

Rewrite the expression.

$\frac{π t}{π} = \frac{π}{4 π}$

Step 2.6.2.2.2

Cancel the common factor of $π$ .

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Step 2.6.2.2.2.1

Cancel the common factor.

$\frac{π t}{π} = \frac{π}{4 π}$

Step 2.6.2.2.2.2

Divide $t$ by $1$ .

$t = \frac{π}{4 π}$

Step 2.6.2.3

Simplify the right side.

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Step 2.6.2.3.1

Cancel the common factor of $π$ .

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Step 2.6.2.3.1.1

Cancel the common factor.

$t = \frac{π}{4 π}$

Step 2.6.2.3.1.2

Rewrite the expression.

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

Step 2.7

Find the period of $\tan (4 π t)$ .

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Step 2.7.1

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

$\frac{π}{| b |}$

Step 2.7.2

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

$\frac{π}{| 4 π |}$

Step 2.7.3

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

$\frac{π}{4 π}$

Step 2.7.4

Cancel the common factor of $π$ .

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Step 2.7.4.1

Cancel the common factor.

$\frac{π}{4 π}$

Step 2.7.4.2

Rewrite the expression.

$\frac{1}{4}$

Step 2.8

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 $n$

Step 2.9

Consolidate the answers.

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

Step 3