Precalculus Examples

$3 \tan (\frac{x}{2})$

Step 1

Set $3 \tan (\frac{x}{2})$ equal to $0$ .

$3 \tan (\frac{x}{2}) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Divide each term in $3 \tan (\frac{x}{2}) = 0$ by $3$ and simplify.

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Step 2.1.1

Divide each term in $3 \tan (\frac{x}{2}) = 0$ by $3$ .

$\frac{3 \tan (\frac{x}{2})}{3} = \frac{0}{3}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $3$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{3 \tan (\frac{x}{2})}{3} = \frac{0}{3}$

Step 2.1.2.1.2

Divide $\tan (\frac{x}{2})$ by $1$ .

$\tan (\frac{x}{2}) = \frac{0}{3}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $3$ .

$\tan (\frac{x}{2}) = 0$

Step 2.2

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

$\frac{x}{2} = \arctan (0)$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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

Step 2.4

Set the numerator equal to zero.

$x = 0$

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.

$\frac{x}{2} = π + 0$

Step 2.6

Solve for $x$ .

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Step 2.6.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π + 0)$

Step 2.6.2

Simplify both sides of the equation.

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Step 2.6.2.1

Simplify the left side.

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Step 2.6.2.1.1

Cancel the common factor of $2$ .

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Step 2.6.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π + 0)$

Step 2.6.2.1.1.2

Rewrite the expression.

$x = 2 (π + 0)$

Step 2.6.2.2

Simplify the right side.

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Step 2.6.2.2.1

Add $π$ and $0$ .

$x = 2 π$

Step 2.7

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

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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 $\frac{1}{2}$ in the formula for period.

$\frac{π}{| \frac{1}{2} |}$

Step 2.7.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{2}}$

Step 2.7.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 2.7.5

Move $2$ to the left of $π$ .

$2 π$

Step 2.8

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

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

Step 2.9

Consolidate the answers.

$x = 2 π n$ , for any integer $n$

Step 3