Trigonometry Examples

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

Step 1

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 2

Simplify the right side.

Tap for more steps...

Step 2.1

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

$3 x = \frac{π}{6}$

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{3}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{3}$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

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

$x = \frac{π}{6 \cdot 3}$

Step 3.3.2.2

Multiply $6$ by $3$ .

$x = \frac{π}{18}$

Step 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.

$3 x = π + \frac{π}{6}$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

Simplify.

Tap for more steps...

Step 5.1.1

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

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

Step 5.1.2

Combine $π$ and $\frac{6}{6}$ .

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

Step 5.1.3

Combine the numerators over the common denominator.

$3 x = \frac{π \cdot 6 + π}{6}$

Step 5.1.4

Add $π \cdot 6$ and $π$ .

Tap for more steps...

Step 5.1.4.1

Reorder $π$ and $6$ .

$3 x = \frac{6 \cdot π + π}{6}$

Step 5.1.4.2

Add $6 \cdot π$ and $π$ .

$3 x = \frac{7 \cdot π}{6}$

Step 5.2

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

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7 \cdot π}{6}}{3}$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.3.2

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

Tap for more steps...

Step 5.2.3.2.1

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

$x = \frac{7 π}{6 \cdot 3}$

Step 5.2.3.2.2

Multiply $6$ by $3$ .

$x = \frac{7 π}{18}$

Step 6

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

Tap for more steps...

Step 6.1

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

$\frac{π}{| b |}$

Step 6.2

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

$\frac{π}{| 3 |}$

Step 6.3

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

$\frac{π}{3}$

Step 7

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 8

Consolidate the answers.

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