Trigonometry Examples

$\tan (\frac{x}{2} + \frac{π}{6}) = - 1$

Step 1

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

$\frac{x}{2} + \frac{π}{6} = \arctan (- 1)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$\frac{x}{2} + \frac{π}{6} = - \frac{π}{4}$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$\frac{x}{2} = - \frac{π}{4} - \frac{π}{6}$

Step 3.2

To write $- \frac{π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{2} = - \frac{π}{4} \cdot \frac{3}{3} - \frac{π}{6}$

Step 3.3

To write $- \frac{π}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{2} = - \frac{π}{4} \cdot \frac{3}{3} - \frac{π}{6} \cdot \frac{2}{2}$

Step 3.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{π}{4}$ by $\frac{3}{3}$ .

$\frac{x}{2} = - \frac{π \cdot 3}{4 \cdot 3} - \frac{π}{6} \cdot \frac{2}{2}$

Step 3.4.2

Multiply $4$ by $3$ .

$\frac{x}{2} = - \frac{π \cdot 3}{12} - \frac{π}{6} \cdot \frac{2}{2}$

Step 3.4.3

Multiply $\frac{π}{6}$ by $\frac{2}{2}$ .

$\frac{x}{2} = - \frac{π \cdot 3}{12} - \frac{π \cdot 2}{6 \cdot 2}$

Step 3.4.4

Multiply $6$ by $2$ .

$\frac{x}{2} = - \frac{π \cdot 3}{12} - \frac{π \cdot 2}{12}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{- π \cdot 3 - π \cdot 2}{12}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $3$ by $- 1$ .

$\frac{x}{2} = \frac{- 3 π - π \cdot 2}{12}$

Step 3.6.2

Multiply $2$ by $- 1$ .

$\frac{x}{2} = \frac{- 3 π - 2 π}{12}$

Step 3.6.3

Subtract $2 π$ from $- 3 π$ .

$\frac{x}{2} = \frac{- 5 π}{12}$

Step 3.7

Move the negative in front of the fraction.

$\frac{x}{2} = - \frac{5 π}{12}$

Step 4

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (- \frac{5 π}{12})$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (- \frac{5 π}{12})$

Step 5.1.1.2

Rewrite the expression.

$x = 2 (- \frac{5 π}{12})$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $2 (- \frac{5 π}{12})$ .

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Move the leading negative in $- \frac{5 π}{12}$ into the numerator.

$x = 2 \frac{- 5 π}{12}$

Step 5.2.1.1.2

Factor $2$ out of $12$ .

$x = 2 \frac{- 5 π}{2 (6)}$

Step 5.2.1.1.3

Cancel the common factor.

$x = 2 \frac{- 5 π}{2 \cdot 6}$

Step 5.2.1.1.4

Rewrite the expression.

$x = \frac{- 5 π}{6}$

Step 5.2.1.2

Move the negative in front of the fraction.

$x = - \frac{5 π}{6}$

Step 6

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.

$\frac{x}{2} + \frac{π}{6} = - \frac{π}{4} - π$

Step 7

Simplify the expression to find the second solution.

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Step 7.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$\frac{x}{2} + \frac{π}{6} = - \frac{π}{4} - π + 2 π$

Step 7.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$\frac{x}{2} + \frac{π}{6} = \frac{3 π}{4}$

Step 7.3

Solve for $x$ .

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Step 7.3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.1.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$\frac{x}{2} = \frac{3 π}{4} - \frac{π}{6}$

Step 7.3.1.2

To write $\frac{3 π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{2} = \frac{3 π}{4} \cdot \frac{3}{3} - \frac{π}{6}$

Step 7.3.1.3

To write $- \frac{π}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{2} = \frac{3 π}{4} \cdot \frac{3}{3} - \frac{π}{6} \cdot \frac{2}{2}$

Step 7.3.1.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.1.4.1

Multiply $\frac{3 π}{4}$ by $\frac{3}{3}$ .

$\frac{x}{2} = \frac{3 π \cdot 3}{4 \cdot 3} - \frac{π}{6} \cdot \frac{2}{2}$

Step 7.3.1.4.2

Multiply $4$ by $3$ .

$\frac{x}{2} = \frac{3 π \cdot 3}{12} - \frac{π}{6} \cdot \frac{2}{2}$

Step 7.3.1.4.3

Multiply $\frac{π}{6}$ by $\frac{2}{2}$ .

$\frac{x}{2} = \frac{3 π \cdot 3}{12} - \frac{π \cdot 2}{6 \cdot 2}$

Step 7.3.1.4.4

Multiply $6$ by $2$ .

$\frac{x}{2} = \frac{3 π \cdot 3}{12} - \frac{π \cdot 2}{12}$

Step 7.3.1.5

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{3 π \cdot 3 - π \cdot 2}{12}$

Step 7.3.1.6

Simplify the numerator.

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Step 7.3.1.6.1

Multiply $3$ by $3$ .

$\frac{x}{2} = \frac{9 π - π \cdot 2}{12}$

Step 7.3.1.6.2

Multiply $2$ by $- 1$ .

$\frac{x}{2} = \frac{9 π - 2 π}{12}$

Step 7.3.1.6.3

Subtract $2 π$ from $9 π$ .

$\frac{x}{2} = \frac{7 π}{12}$

Step 7.3.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{7 π}{12}$

Step 7.3.3

Simplify both sides of the equation.

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Step 7.3.3.1

Simplify the left side.

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Step 7.3.3.1.1

Cancel the common factor of $2$ .

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Step 7.3.3.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{7 π}{12}$

Step 7.3.3.1.1.2

Rewrite the expression.

$x = 2 \frac{7 π}{12}$

Step 7.3.3.2

Simplify the right side.

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Step 7.3.3.2.1

Cancel the common factor of $2$ .

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Step 7.3.3.2.1.1

Factor $2$ out of $12$ .

$x = 2 \frac{7 π}{2 (6)}$

Step 7.3.3.2.1.2

Cancel the common factor.

$x = 2 \frac{7 π}{2 \cdot 6}$

Step 7.3.3.2.1.3

Rewrite the expression.

$x = \frac{7 π}{6}$

Step 8

Find the period of $\tan (\frac{x}{2} + \frac{π}{6})$ .

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Step 8.1

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

$\frac{π}{| b |}$

Step 8.2

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

$\frac{π}{| \frac{1}{2} |}$

Step 8.3

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

$\frac{π}{\frac{1}{2}}$

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 8.5

Move $2$ to the left of $π$ .

$2 π$

Step 9

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

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Step 9.1

Add $2 π$ to $- \frac{5 π}{6}$ to find the positive angle.

$- \frac{5 π}{6} + 2 π$

Step 9.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 π \cdot \frac{6}{6} - \frac{5 π}{6}$

Step 9.3

Combine fractions.

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Step 9.3.1

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{5 π}{6}$

Step 9.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - 5 π}{6}$

Step 9.4

Simplify the numerator.

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Step 9.4.1

Multiply $6$ by $2$ .

$\frac{12 π - 5 π}{6}$

Step 9.4.2

Subtract $5 π$ from $12 π$ .

$\frac{7 π}{6}$

Step 9.5

List the new angles.

$x = \frac{7 π}{6}$

Step 10

The period of the $\tan (\frac{x}{2} + \frac{π}{6})$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{7 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n$ , for any integer $n$

Step 11

Consolidate the answers.

$x = \frac{7 π}{6} + 2 π n$ , for any integer $n$