Precalculus Examples

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

Step 1

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

$\frac{x}{2} = \arctan (- \frac{\sqrt{3}}{3})$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{x}{2} = - \frac{π}{6}$

Step 3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (- \frac{π}{6})$

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 $2$ .

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Step 4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (- \frac{π}{6})$

Step 4.1.1.2

Rewrite the expression.

$x = 2 (- \frac{π}{6})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 (- \frac{π}{6})$ .

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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

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$x = 2 \frac{- π}{6}$

Step 4.2.1.1.2

Factor $2$ out of $6$ .

$x = 2 \frac{- π}{2 (3)}$

Step 4.2.1.1.3

Cancel the common factor.

$x = 2 \frac{- π}{2 \cdot 3}$

Step 4.2.1.1.4

Rewrite the expression.

$x = \frac{- π}{3}$

Step 4.2.1.2

Move the negative in front of the fraction.

$x = - \frac{π}{3}$

Step 5

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$\frac{x}{2} = - \frac{π}{6} - π$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Add $2 π$ to $- \frac{π}{6} - π$ .

$\frac{x}{2} = - \frac{π}{6} - π + 2 π$

Step 6.2

The resulting angle of $\frac{5 π}{6}$ is positive and coterminal with $- \frac{π}{6} - π$ .

$\frac{x}{2} = \frac{5 π}{6}$

Step 6.3

Solve for $x$ .

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Step 6.3.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{5 π}{6}$

Step 6.3.2

Simplify both sides of the equation.

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Step 6.3.2.1

Simplify the left side.

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Step 6.3.2.1.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{5 π}{6}$

Step 6.3.2.1.1.2

Rewrite the expression.

$x = 2 \frac{5 π}{6}$

Step 6.3.2.2

Simplify the right side.

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Step 6.3.2.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.1

Factor $2$ out of $6$ .

$x = 2 \frac{5 π}{2 (3)}$

Step 6.3.2.2.1.2

Cancel the common factor.

$x = 2 \frac{5 π}{2 \cdot 3}$

Step 6.3.2.2.1.3

Rewrite the expression.

$x = \frac{5 π}{3}$

Step 7

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

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

$\frac{π}{| \frac{1}{2} |}$

Step 7.3

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

$\frac{π}{\frac{1}{2}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 7.5

Move $2$ to the left of $π$ .

$2 π$

Step 8

Add $2 π$ to every negative angle to get positive angles.

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Step 8.1

Add $2 π$ to $- \frac{π}{3}$ to find the positive angle.

$- \frac{π}{3} + 2 π$

Step 8.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 8.3

Combine fractions.

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Step 8.3.1

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 8.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{3}$

Step 8.4

Simplify the numerator.

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Step 8.4.1

Multiply $3$ by $2$ .

$\frac{6 π - π}{3}$

Step 8.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{3}$

Step 8.5

List the new angles.

$x = \frac{5 π}{3}$

Step 9

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

$x = \frac{5 π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 10

Consolidate the answers.

$x = \frac{5 π}{3} + 2 π n$ , for any integer $n$