Solve for x 3tan(x)^2=tan(x)

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

Step 1

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

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

Step 2

Factor $\tan (x)$ out of $3 \tan^{2} (x) - \tan (x)$ .

Factor $\tan (x)$ out of $3 \tan^{2} (x)$ .

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

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

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

Factor $\tan (x)$ out of $\tan (x) (3 \tan (x)) + \tan (x) \cdot - 1$ .

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

Step 3

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

$\tan (x) = 0$

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

Step 4

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

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

$\tan (x) = 0$

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

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

$x = \arctan (0)$

Simplify the right side.

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

$x = 0$

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.

$x = π + 0$

Add $π$ and $0$ .

$x = π$

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

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

$\frac{π}{| b |}$

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

$\frac{π}{| 1 |}$

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

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

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

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

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

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

Add $1$ to both sides of the equation.

$3 \tan (x) = 1$

Divide each term in $3 \tan (x) = 1$ by $3$ and simplify.

Divide each term in $3 \tan (x) = 1$ by $3$ .

$\frac{3 \tan (x)}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 \tan (x)}{3} = \frac{1}{3}$

Divide $\tan (x)$ by $1$ .

$\tan (x) = \frac{1}{3}$

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

$x = \arctan (\frac{1}{3})$

Simplify the right side.

Evaluate $\arctan (\frac{1}{3})$ .

$x = 0.32175055$

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.

$x = (3.14159265) + 0.32175055$

Solve for $x$ .

Remove parentheses.

$x = 3.14159265 + 0.32175055$

Remove parentheses.

$x = (3.14159265) + 0.32175055$

Add $3.14159265$ and $0.32175055$ .

$x = 3.4633432$

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

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

$\frac{π}{| b |}$

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

$\frac{π}{| 1 |}$

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

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

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

$x = 0.32175055 + π n, 3.4633432 + π n$ , for any integer $n$

Step 6

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

$x = π n, π + π n, 0.32175055 + π n, 3.4633432 + π n$ , for any integer $n$

Step 7

Consolidate the answers.

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

$x = π n, 0.32175055 + π n, 3.4633432 + π n$ , for any integer $n$

Consolidate $0.32175055 + π n$ and $3.4633432 + π n$ to $0.32175055 + π n$ .

$x = π n, 0.32175055 + π n$ , for any integer $n$

Solve for x 3tan(x)^2=tan(x)

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