Precalculus Examples

$\sqrt{3} \tan (x - \frac{π}{9}) - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$\sqrt{3} \tan (x - \frac{π}{9}) = 1$

Step 2

Divide each term in $\sqrt{3} \tan (x - \frac{π}{9}) = 1$ by $\sqrt{3}$ and simplify.

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Step 2.1

Divide each term in $\sqrt{3} \tan (x - \frac{π}{9}) = 1$ by $\sqrt{3}$ .

$\frac{\sqrt{3} \tan (x - \frac{π}{9})}{\sqrt{3}} = \frac{1}{\sqrt{3}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $\sqrt{3}$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{\sqrt{3} \tan (x - \frac{π}{9})}{\sqrt{3}} = \frac{1}{\sqrt{3}}$

Step 2.2.1.2

Divide $\tan (x - \frac{π}{9})$ by $1$ .

$\tan (x - \frac{π}{9}) = \frac{1}{\sqrt{3}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\tan (x - \frac{π}{9}) = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.3.2

Combine and simplify the denominator.

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Step 2.3.2.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\tan (x - \frac{π}{9}) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.3.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\tan (x - \frac{π}{9}) = \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.3.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\tan (x - \frac{π}{9}) = \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.3.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\tan (x - \frac{π}{9}) = \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.3.2.5

Add $1$ and $1$ .

$\tan (x - \frac{π}{9}) = \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.3.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\tan (x - \frac{π}{9}) = \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.3.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\tan (x - \frac{π}{9}) = \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.3.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\tan (x - \frac{π}{9}) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.2.6.4

Cancel the common factor of $2$ .

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Step 2.3.2.6.4.1

Cancel the common factor.

$\tan (x - \frac{π}{9}) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.2.6.4.2

Rewrite the expression.

$\tan (x - \frac{π}{9}) = \frac{\sqrt{3}}{3^{1}}$

Step 2.3.2.6.5

Evaluate the exponent.

$\tan (x - \frac{π}{9}) = \frac{\sqrt{3}}{3}$

Step 3

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

$x - \frac{π}{9} = \arctan (\frac{\sqrt{3}}{3})$

Step 4

Simplify the right side.

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Step 4.1

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

$x - \frac{π}{9} = \frac{π}{6}$

Step 5

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

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Step 5.1

Add $\frac{π}{9}$ to both sides of the equation.

$x = \frac{π}{6} + \frac{π}{9}$

Step 5.2

To write $\frac{π}{6}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{π}{6} \cdot \frac{3}{3} + \frac{π}{9}$

Step 5.3

To write $\frac{π}{9}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{π}{6} \cdot \frac{3}{3} + \frac{π}{9} \cdot \frac{2}{2}$

Step 5.4

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.1

Multiply $\frac{π}{6}$ by $\frac{3}{3}$ .

$x = \frac{π \cdot 3}{6 \cdot 3} + \frac{π}{9} \cdot \frac{2}{2}$

Step 5.4.2

Multiply $6$ by $3$ .

$x = \frac{π \cdot 3}{18} + \frac{π}{9} \cdot \frac{2}{2}$

Step 5.4.3

Multiply $\frac{π}{9}$ by $\frac{2}{2}$ .

$x = \frac{π \cdot 3}{18} + \frac{π \cdot 2}{9 \cdot 2}$

Step 5.4.4

Multiply $9$ by $2$ .

$x = \frac{π \cdot 3}{18} + \frac{π \cdot 2}{18}$

Step 5.5

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 + π \cdot 2}{18}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π + π \cdot 2}{18}$

Step 5.6.2

Move $2$ to the left of $π$ .

$x = \frac{3 π + 2 \cdot π}{18}$

Step 5.6.3

Add $3 π$ and $2 π$ .

$x = \frac{5 π}{18}$

Step 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.

$x - \frac{π}{9} = π + \frac{π}{6}$

Step 7

Solve for $x$ .

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Step 7.1

Simplify $π + \frac{π}{6}$ .

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Step 7.1.1

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

$x - \frac{π}{9} = π \cdot \frac{6}{6} + \frac{π}{6}$

Step 7.1.2

Combine fractions.

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Step 7.1.2.1

Combine $π$ and $\frac{6}{6}$ .

$x - \frac{π}{9} = \frac{π \cdot 6}{6} + \frac{π}{6}$

Step 7.1.2.2

Combine the numerators over the common denominator.

$x - \frac{π}{9} = \frac{π \cdot 6 + π}{6}$

Step 7.1.3

Simplify the numerator.

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Step 7.1.3.1

Move $6$ to the left of $π$ .

$x - \frac{π}{9} = \frac{6 \cdot π + π}{6}$

Step 7.1.3.2

Add $6 π$ and $π$ .

$x - \frac{π}{9} = \frac{7 π}{6}$

Step 7.2

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

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Step 7.2.1

Add $\frac{π}{9}$ to both sides of the equation.

$x = \frac{7 π}{6} + \frac{π}{9}$

Step 7.2.2

To write $\frac{7 π}{6}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{7 π}{6} \cdot \frac{3}{3} + \frac{π}{9}$

Step 7.2.3

To write $\frac{π}{9}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{7 π}{6} \cdot \frac{3}{3} + \frac{π}{9} \cdot \frac{2}{2}$

Step 7.2.4

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.4.1

Multiply $\frac{7 π}{6}$ by $\frac{3}{3}$ .

$x = \frac{7 π \cdot 3}{6 \cdot 3} + \frac{π}{9} \cdot \frac{2}{2}$

Step 7.2.4.2

Multiply $6$ by $3$ .

$x = \frac{7 π \cdot 3}{18} + \frac{π}{9} \cdot \frac{2}{2}$

Step 7.2.4.3

Multiply $\frac{π}{9}$ by $\frac{2}{2}$ .

$x = \frac{7 π \cdot 3}{18} + \frac{π \cdot 2}{9 \cdot 2}$

Step 7.2.4.4

Multiply $9$ by $2$ .

$x = \frac{7 π \cdot 3}{18} + \frac{π \cdot 2}{18}$

Step 7.2.5

Combine the numerators over the common denominator.

$x = \frac{7 π \cdot 3 + π \cdot 2}{18}$

Step 7.2.6

Simplify the numerator.

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Step 7.2.6.1

Multiply $3$ by $7$ .

$x = \frac{21 π + π \cdot 2}{18}$

Step 7.2.6.2

Move $2$ to the left of $π$ .

$x = \frac{21 π + 2 \cdot π}{18}$

Step 7.2.6.3

Add $21 π$ and $2 π$ .

$x = \frac{23 π}{18}$

Step 8

Find the period of $\tan (x - \frac{π}{9})$ .

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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 $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 8.4

Divide $π$ by $1$ .

$π$

Step 9

The period of the $\tan (x - \frac{π}{9})$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{5 π}{18} + π n, \frac{23 π}{18} + π n$ , for any integer $n$

Step 10

Consolidate the answers.

$x = \frac{5 π}{18} + π n$ , for any integer $n$