Precalculus Examples

$\tan^{2} (3 x) = \sqrt{1}$

Step 1

Subtract $\sqrt{1}$ from both sides of the equation.

$\tan^{2} (3 x) - \sqrt{1} = 0$

Step 2

Simplify each term.

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Step 2.1

Any root of $1$ is $1$ .

$\tan^{2} (3 x) - 1 \cdot 1 = 0$

Step 2.2

Multiply $- 1$ by $1$ .

$\tan^{2} (3 x) - 1 = 0$

Step 3

Rewrite $1$ as $1^{2}$ .

$\tan^{2} (3 x) - 1^{2} = 0$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \tan (3 x)$ and $b = 1$ .

$(\tan (3 x) + 1) (\tan (3 x) - 1) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\tan (3 x) + 1 = 0$

$\tan (3 x) - 1 = 0$

Step 6

Set $\tan (3 x) + 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $\tan (3 x) + 1$ equal to $0$ .

$\tan (3 x) + 1 = 0$

Step 6.2

Solve $\tan (3 x) + 1 = 0$ for $x$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$\tan (3 x) = - 1$

Step 6.2.2

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

$3 x = \arctan (- 1)$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

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

$3 x = - \frac{π}{4}$

Step 6.2.4

Divide each term in $3 x = - \frac{π}{4}$ by $3$ and simplify.

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Step 6.2.4.1

Divide each term in $3 x = - \frac{π}{4}$ by $3$ .

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

Step 6.2.4.2

Simplify the left side.

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Step 6.2.4.2.1

Cancel the common factor of $3$ .

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Step 6.2.4.2.1.1

Cancel the common factor.

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

Step 6.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{4}}{3}$

Step 6.2.4.3

Simplify the right side.

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Step 6.2.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{4} \cdot \frac{1}{3}$

Step 6.2.4.3.2

Multiply $- \frac{π}{4} \cdot \frac{1}{3}$ .

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Step 6.2.4.3.2.1

Multiply $\frac{1}{3}$ by $\frac{π}{4}$ .

$x = - \frac{π}{3 \cdot 4}$

Step 6.2.4.3.2.2

Multiply $3$ by $4$ .

$x = - \frac{π}{12}$

Step 6.2.5

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.

$3 x = - \frac{π}{4} - π$

Step 6.2.6

Simplify the expression to find the second solution.

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Step 6.2.6.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$3 x = - \frac{π}{4} - π + 2 π$

Step 6.2.6.2

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

$3 x = \frac{3 π}{4}$

Step 6.2.6.3

Divide each term in $3 x = \frac{3 π}{4}$ by $3$ and simplify.

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Step 6.2.6.3.1

Divide each term in $3 x = \frac{3 π}{4}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{3 π}{4}}{3}$

Step 6.2.6.3.2

Simplify the left side.

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Step 6.2.6.3.2.1

Cancel the common factor of $3$ .

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Step 6.2.6.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{3 π}{4}}{3}$

Step 6.2.6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{4}}{3}$

Step 6.2.6.3.3

Simplify the right side.

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Step 6.2.6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{4} \cdot \frac{1}{3}$

Step 6.2.6.3.3.2

Cancel the common factor of $3$ .

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Step 6.2.6.3.3.2.1

Factor $3$ out of $3 π$ .

$x = \frac{3 (π)}{4} \cdot \frac{1}{3}$

Step 6.2.6.3.3.2.2

Cancel the common factor.

$x = \frac{3 π}{4} \cdot \frac{1}{3}$

Step 6.2.6.3.3.2.3

Rewrite the expression.

$x = \frac{π}{4}$

Step 6.2.7

Find the period of $\tan (3 x)$ .

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Step 6.2.7.1

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

$\frac{π}{| b |}$

Step 6.2.7.2

Replace $b$ with $3$ in the formula for period.

$\frac{π}{| 3 |}$

Step 6.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{π}{3}$

Step 6.2.8

Add $\frac{π}{3}$ to every negative angle to get positive angles.

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Step 6.2.8.1

Add $\frac{π}{3}$ to $- \frac{π}{12}$ to find the positive angle.

$- \frac{π}{12} + \frac{π}{3}$

Step 6.2.8.2

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

$\frac{π}{3} \cdot \frac{4}{4} - \frac{π}{12}$

Step 6.2.8.3

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

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Step 6.2.8.3.1

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

$\frac{π \cdot 4}{3 \cdot 4} - \frac{π}{12}$

Step 6.2.8.3.2

Multiply $3$ by $4$ .

$\frac{π \cdot 4}{12} - \frac{π}{12}$

Step 6.2.8.4

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{12}$

Step 6.2.8.5

Simplify the numerator.

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Step 6.2.8.5.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{12}$

Step 6.2.8.5.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{12}$

Step 6.2.8.6

Cancel the common factor of $3$ and $12$ .

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Step 6.2.8.6.1

Factor $3$ out of $3 π$ .

$\frac{3 (π)}{12}$

Step 6.2.8.6.2

Cancel the common factors.

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Step 6.2.8.6.2.1

Factor $3$ out of $12$ .

$\frac{3 π}{3 \cdot 4}$

Step 6.2.8.6.2.2

Cancel the common factor.

$\frac{3 π}{3 \cdot 4}$

Step 6.2.8.6.2.3

Rewrite the expression.

$\frac{π}{4}$

Step 6.2.8.7

List the new angles.

$x = \frac{π}{4}$

Step 6.2.9

The period of the $\tan (3 x)$ function is $\frac{π}{3}$ so values will repeat every $\frac{π}{3}$ radians in both directions.

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

Step 7

Set $\tan (3 x) - 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $\tan (3 x) - 1$ equal to $0$ .

$\tan (3 x) - 1 = 0$

Step 7.2

Solve $\tan (3 x) - 1 = 0$ for $x$ .

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Step 7.2.1

Add $1$ to both sides of the equation.

$\tan (3 x) = 1$

Step 7.2.2

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

$3 x = \arctan (1)$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

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

$3 x = \frac{π}{4}$

Step 7.2.4

Divide each term in $3 x = \frac{π}{4}$ by $3$ and simplify.

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Step 7.2.4.1

Divide each term in $3 x = \frac{π}{4}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{π}{4}}{3}$

Step 7.2.4.2

Simplify the left side.

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Step 7.2.4.2.1

Cancel the common factor of $3$ .

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Step 7.2.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{π}{4}}{3}$

Step 7.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{4}}{3}$

Step 7.2.4.3

Simplify the right side.

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Step 7.2.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{4} \cdot \frac{1}{3}$

Step 7.2.4.3.2

Multiply $\frac{π}{4} \cdot \frac{1}{3}$ .

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Step 7.2.4.3.2.1

Multiply $\frac{π}{4}$ by $\frac{1}{3}$ .

$x = \frac{π}{4 \cdot 3}$

Step 7.2.4.3.2.2

Multiply $4$ by $3$ .

$x = \frac{π}{12}$

Step 7.2.5

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.

$3 x = π + \frac{π}{4}$

Step 7.2.6

Solve for $x$ .

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Step 7.2.6.1

Simplify.

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Step 7.2.6.1.1

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

$3 x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 7.2.6.1.2

Combine $π$ and $\frac{4}{4}$ .

$3 x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 7.2.6.1.3

Combine the numerators over the common denominator.

$3 x = \frac{π \cdot 4 + π}{4}$

Step 7.2.6.1.4

Add $π \cdot 4$ and $π$ .

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Step 7.2.6.1.4.1

Reorder $π$ and $4$ .

$3 x = \frac{4 \cdot π + π}{4}$

Step 7.2.6.1.4.2

Add $4 \cdot π$ and $π$ .

$3 x = \frac{5 \cdot π}{4}$

Step 7.2.6.2

Divide each term in $3 x = \frac{5 \cdot π}{4}$ by $3$ and simplify.

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Step 7.2.6.2.1

Divide each term in $3 x = \frac{5 \cdot π}{4}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{5 \cdot π}{4}}{3}$

Step 7.2.6.2.2

Simplify the left side.

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Step 7.2.6.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.6.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{5 \cdot π}{4}}{3}$

Step 7.2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 \cdot π}{4}}{3}$

Step 7.2.6.2.3

Simplify the right side.

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Step 7.2.6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 \cdot π}{4} \cdot \frac{1}{3}$

Step 7.2.6.2.3.2

Multiply $\frac{5 π}{4} \cdot \frac{1}{3}$ .

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Step 7.2.6.2.3.2.1

Multiply $\frac{5 π}{4}$ by $\frac{1}{3}$ .

$x = \frac{5 π}{4 \cdot 3}$

Step 7.2.6.2.3.2.2

Multiply $4$ by $3$ .

$x = \frac{5 π}{12}$

Step 7.2.7

Find the period of $\tan (3 x)$ .

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Step 7.2.7.1

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

$\frac{π}{| b |}$

Step 7.2.7.2

Replace $b$ with $3$ in the formula for period.

$\frac{π}{| 3 |}$

Step 7.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{π}{3}$

Step 7.2.8

The period of the $\tan (3 x)$ function is $\frac{π}{3}$ so values will repeat every $\frac{π}{3}$ radians in both directions.

$x = \frac{π}{12} + \frac{π n}{3}, \frac{5 π}{12} + \frac{π n}{3}$ , for any integer $n$

Step 8

The final solution is all the values that make $(\tan (3 x) + 1) (\tan (3 x) - 1) = 0$ true.

$x = \frac{π}{4} + \frac{π n}{3}, \frac{π}{12} + \frac{π n}{3}, \frac{5 π}{12} + \frac{π n}{3}$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{π}{12} + \frac{π n}{6}$ , for any integer $n$