Precalculus Examples

$\tan^{3} (x) = \tan (x)$

Step 1

Subtract $\tan (x)$ from both sides of the equation.

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

Step 2

Factor $\tan^{3} (x) - \tan (x)$ .

Tap for more steps...

Step 2.1

Factor $\tan (x)$ out of $\tan^{3} (x) - \tan (x)$ .

Tap for more steps...

Step 2.1.1

Factor $\tan (x)$ out of $\tan^{3} (x)$ .

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

Step 2.1.2

Factor $\tan (x)$ out of $- \tan (x)$ .

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

Step 2.1.3

Factor $\tan (x)$ out of $\tan (x) \tan^{2} (x) + \tan (x) \cdot - 1$ .

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

Step 2.2

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

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

Step 2.3

Factor.

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

Remove unnecessary parentheses.

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

Step 3

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

$\tan (x) = 0$

$\tan (x) + 1 = 0$

$\tan (x) - 1 = 0$

Step 4

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

Tap for more steps...

Step 4.1

Set $\tan (x)$ equal to $0$ .

$\tan (x) = 0$

Step 4.2

Solve $\tan (x) = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

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

$x = \arctan (0)$

Step 4.2.2

Simplify the right side.

Tap for more steps...

Step 4.2.2.1

The exact value of $\arctan (0)$ is $0$ .

$x = 0$

Step 4.2.3

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.

$x = π + 0$

Step 4.2.4

Add $π$ and $0$ .

$x = π$

Step 4.2.5

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

Tap for more steps...

Step 4.2.5.1

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

$\frac{π}{| b |}$

Step 4.2.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 4.2.5.3

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

$\frac{π}{1}$

Step 4.2.5.4

Divide $π$ by $1$ .

$π$

Step 4.2.6

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

$x = π n, π + π n$ , for any integer $n$

Step 5

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

Tap for more steps...

Step 5.1

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

$\tan (x) + 1 = 0$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

Step 5.2.2

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

$x = \arctan (- 1)$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

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

$x = - \frac{π}{4}$

Step 5.2.4

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.

$x = - \frac{π}{4} - π$

Step 5.2.5

Simplify the expression to find the second solution.

Tap for more steps...

Step 5.2.5.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 5.2.5.2

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

$x = \frac{3 π}{4}$

Step 5.2.6

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

Tap for more steps...

Step 5.2.6.1

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

$\frac{π}{| b |}$

Step 5.2.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 5.2.6.3

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

$\frac{π}{1}$

Step 5.2.6.4

Divide $π$ by $1$ .

$π$

Step 5.2.7

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

Tap for more steps...

Step 5.2.7.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 5.2.7.2

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

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 5.2.7.3

Combine fractions.

Tap for more steps...

Step 5.2.7.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 5.2.7.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 5.2.7.4

Simplify the numerator.

Tap for more steps...

Step 5.2.7.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 5.2.7.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 5.2.7.5

List the new angles.

$x = \frac{3 π}{4}$

Step 5.2.8

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

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

Step 6

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

Tap for more steps...

Step 6.1

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

$\tan (x) - 1 = 0$

Step 6.2

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

Tap for more steps...

Step 6.2.1

Add $1$ to both sides of the equation.

$\tan (x) = 1$

Step 6.2.2

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

$x = \arctan (1)$

Step 6.2.3

Simplify the right side.

Tap for more steps...

Step 6.2.3.1

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

$x = \frac{π}{4}$

Step 6.2.4

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.

$x = π + \frac{π}{4}$

Step 6.2.5

Simplify $π + \frac{π}{4}$ .

Tap for more steps...

Step 6.2.5.1

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

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

Step 6.2.5.2

Combine fractions.

Tap for more steps...

Step 6.2.5.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 6.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 6.2.5.3

Simplify the numerator.

Tap for more steps...

Step 6.2.5.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 6.2.5.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 6.2.6

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

Tap for more steps...

Step 6.2.6.1

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

$\frac{π}{| b |}$

Step 6.2.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 6.2.6.3

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

$\frac{π}{1}$

Step 6.2.6.4

Divide $π$ by $1$ .

$π$

Step 6.2.7

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

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

Step 7

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

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

Step 8

Consolidate the answers.

Tap for more steps...

Step 8.1

Consolidate $π n$ and $π + π n$ to $π n$ .

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

Step 8.2

Consolidate $\frac{3 π}{4} + π n$ and $\frac{π}{4} + π n$ to $\frac{π}{4} + \frac{π n}{2}$ .

$x = π n, \frac{π}{4} + \frac{π n}{2}, \frac{5 π}{4} + π n$ , for any integer $n$

Step 8.3

Consolidate $\frac{π}{4} + \frac{π n}{2}$ and $\frac{5 π}{4} + π n$ to $\frac{π}{4} + \frac{π n}{2}$ .

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