Precalculus Examples

$\tan (\frac{x}{4}) = \frac{\sqrt{3}}{3}$

Step 1

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

$\frac{x}{4} = \arctan (\frac{\sqrt{3}}{3})$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arctan (\frac{\sqrt{3}}{3})$ is $\frac{π}{6}$ .

$\frac{x}{4} = \frac{π}{6}$

Step 3

Multiply both sides of the equation by $4$ .

$4 \frac{x}{4} = 4 \frac{π}{6}$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $4$ .

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Step 4.1.1.1

Cancel the common factor.

$4 \frac{x}{4} = 4 \frac{π}{6}$

Step 4.1.1.2

Rewrite the expression.

$x = 4 \frac{π}{6}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $4 \frac{π}{6}$ .

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Step 4.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1

Factor $2$ out of $4$ .

$x = 2 (2) \frac{π}{6}$

Step 4.2.1.1.2

Factor $2$ out of $6$ .

$x = 2 \cdot 2 \frac{π}{2 \cdot 3}$

Step 4.2.1.1.3

Cancel the common factor.

$x = 2 \cdot 2 \frac{π}{2 \cdot 3}$

Step 4.2.1.1.4

Rewrite the expression.

$x = 2 \frac{π}{3}$

Step 4.2.1.2

Combine $2$ and $\frac{π}{3}$ .

$x = \frac{2 π}{3}$

Step 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.

$\frac{x}{4} = π + \frac{π}{6}$

Step 6

Solve for $x$ .

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Step 6.1

Multiply both sides of the equation by $4$ .

$4 \frac{x}{4} = 4 (π + \frac{π}{6})$

Step 6.2

Simplify both sides of the equation.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Cancel the common factor of $4$ .

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Step 6.2.1.1.1

Cancel the common factor.

$4 \frac{x}{4} = 4 (π + \frac{π}{6})$

Step 6.2.1.1.2

Rewrite the expression.

$x = 4 (π + \frac{π}{6})$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $4 (π + \frac{π}{6})$ .

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Step 6.2.2.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = 4 (π \cdot \frac{6}{6} + \frac{π}{6})$

Step 6.2.2.1.2

Simplify terms.

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Step 6.2.2.1.2.1

Combine $π$ and $\frac{6}{6}$ .

$x = 4 (\frac{π \cdot 6}{6} + \frac{π}{6})$

Step 6.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 4 \frac{π \cdot 6 + π}{6}$

Step 6.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 6.2.2.1.2.3.1

Factor $2$ out of $4$ .

$x = 2 (2) \frac{π \cdot 6 + π}{6}$

Step 6.2.2.1.2.3.2

Factor $2$ out of $6$ .

$x = 2 \cdot 2 \frac{π \cdot 6 + π}{2 \cdot 3}$

Step 6.2.2.1.2.3.3

Cancel the common factor.

$x = 2 \cdot 2 \frac{π \cdot 6 + π}{2 \cdot 3}$

Step 6.2.2.1.2.3.4

Rewrite the expression.

$x = 2 \frac{π \cdot 6 + π}{3}$

Step 6.2.2.1.2.4

Combine $2$ and $\frac{π \cdot 6 + π}{3}$ .

$x = \frac{2 (π \cdot 6 + π)}{3}$

Step 6.2.2.1.3

Simplify the numerator.

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Step 6.2.2.1.3.1

Move $6$ to the left of $π$ .

$x = \frac{2 (6 \cdot π + π)}{3}$

Step 6.2.2.1.3.2

Add $6 π$ and $π$ .

$x = \frac{2 \cdot 7 π}{3}$

Step 6.2.2.1.3.3

Multiply $2$ by $7$ .

$x = \frac{14 π}{3}$

Step 7

Find the period of $\tan (\frac{x}{4})$ .

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Step 7.1

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

$\frac{π}{| b |}$

Step 7.2

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{π}{| \frac{1}{4} |}$

Step 7.3

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{4}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 4$

Step 7.5

Move $4$ to the left of $π$ .

$4 π$

Step 8

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

$x = \frac{2 π}{3} + 4 π n, \frac{14 π}{3} + 4 π n$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{2 π}{3} + 4 π n$ , for any integer $n$