Find All Complex Solutions 3tan(3x) = square root of 3

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

Step 1

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

Divide each term in $3 \tan (3 x) = \sqrt{3}$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $\tan (3 x)$ by $1$ .

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

Step 2

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

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

Step 3

Simplify the right side.

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

$3 x = \frac{π}{6}$

Step 4

Divide each term in $3 x = \frac{π}{6}$ by $3$ and simplify.

Divide each term in $3 x = \frac{π}{6}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{π}{6}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{π}{6}}{3}$

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{3}$

Multiply $\frac{π}{6} \cdot \frac{1}{3}$ .

Multiply $\frac{π}{6}$ by $\frac{1}{3}$ .

$x = \frac{π}{6 \cdot 3}$

Multiply $6$ by $3$ .

$x = \frac{π}{18}$

Step 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.

$3 x = π + \frac{π}{6}$

Step 6

Solve for $x$ .

Simplify.

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

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

Combine $π$ and $\frac{6}{6}$ .

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

Combine the numerators over the common denominator.

$3 x = \frac{π \cdot 6 + π}{6}$

Add $π \cdot 6$ and $π$ .

Reorder $π$ and $6$ .

$3 x = \frac{6 \cdot π + π}{6}$

Add $6 \cdot π$ and $π$ .

$3 x = \frac{7 \cdot π}{6}$

Divide each term in $3 x = \frac{7 \cdot π}{6}$ by $3$ and simplify.

Divide each term in $3 x = \frac{7 \cdot π}{6}$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{\frac{7 \cdot π}{6}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 \cdot π}{6} \cdot \frac{1}{3}$

Multiply $\frac{7 π}{6} \cdot \frac{1}{3}$ .

Multiply $\frac{7 π}{6}$ by $\frac{1}{3}$ .

$x = \frac{7 π}{6 \cdot 3}$

Multiply $6$ by $3$ .

$x = \frac{7 π}{18}$

Step 7

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

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

$\frac{π}{| b |}$

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

$\frac{π}{| 3 |}$

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

$\frac{π}{3}$

Step 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.

$x = \frac{π}{18} + \frac{π n}{3}, \frac{7 π}{18} + \frac{π n}{3}$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{π}{18} + \frac{π n}{3}$ , for any integer $n$

Find All Complex Solutions 3tan(3x) = square root of 3

Related Questions

Support LCMGCF.com by disabling your adblocker.