Algebra Examples

$f (x) = \tan (\frac{π}{2} x)$

Step 1

Set $\tan (\frac{π}{2} x)$ equal to $0$ .

$\tan (\frac{π}{2} x) = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$\frac{π}{2} x = \arctan (0)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Combine $\frac{π}{2}$ and $x$ .

$\frac{π x}{2} = \arctan (0)$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$\frac{π x}{2} = 0$

Step 2.4

Set the numerator equal to zero.

$π x = 0$

Step 2.5

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

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Step 2.5.1

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

$\frac{π x}{π} = \frac{0}{π}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $π$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{π}$

Step 2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{π}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Divide $0$ by $π$ .

$x = 0$

Step 2.6

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.

$\frac{π x}{2} = π + 0$

Step 2.7

Solve for $x$ .

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Step 2.7.1

Multiply both sides of the equation by $\frac{2}{π}$ .

$\frac{2}{π} \cdot \frac{π x}{2} = \frac{2}{π} (π + 0)$

Step 2.7.2

Simplify both sides of the equation.

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Step 2.7.2.1

Simplify the left side.

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Step 2.7.2.1.1

Simplify $\frac{2}{π} \cdot \frac{π x}{2}$ .

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Step 2.7.2.1.1.1

Cancel the common factor of $2$ .

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Step 2.7.2.1.1.1.1

Cancel the common factor.

$\frac{2}{π} \cdot \frac{π x}{2} = \frac{2}{π} (π + 0)$

Step 2.7.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{2}{π} (π + 0)$

Step 2.7.2.1.1.2

Cancel the common factor of $π$ .

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Step 2.7.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{2}{π} (π + 0)$

Step 2.7.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{2}{π} (π + 0)$

Step 2.7.2.1.1.2.3

Rewrite the expression.

$x = \frac{2}{π} (π + 0)$

Step 2.7.2.2

Simplify the right side.

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Step 2.7.2.2.1

Simplify $\frac{2}{π} (π + 0)$ .

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Step 2.7.2.2.1.1

Add $π$ and $0$ .

$x = \frac{2}{π} π$

Step 2.7.2.2.1.2

Cancel the common factor of $π$ .

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Step 2.7.2.2.1.2.1

Cancel the common factor.

$x = \frac{2}{π} π$

Step 2.7.2.2.1.2.2

Rewrite the expression.

$x = 2$

Step 2.8

Find the period of $\tan (\frac{π}{2} x)$ .

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Step 2.8.1

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

$\frac{π}{| b |}$

Step 2.8.2

Replace $b$ with $\frac{π}{2}$ in the formula for period.

$\frac{π}{| \frac{π}{2} |}$

Step 2.8.3

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{π}{\frac{π}{2}}$

Step 2.8.4

Multiply the numerator by the reciprocal of the denominator.

$π \frac{2}{π}$

Step 2.8.5

Cancel the common factor of $π$ .

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Step 2.8.5.1

Cancel the common factor.

$π \frac{2}{π}$

Step 2.8.5.2

Rewrite the expression.

$2$

Step 2.9

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

$x = 2 n, 2 + 2 n$ , for any integer $n$

Step 2.10

Consolidate the answers.

$x = 2 n$ , for any integer $n$

Step 3