Trigonometry Examples

$\tan (θ) = \frac{\sqrt{3}}{3}$

Step 1

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $3$ .

$\tan (θ) \cdot \frac{3}{3} = \frac{\sqrt{3}}{3}$

Step 2

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $3$ .

$\tan (θ) \cdot 3$

Step 3

Move $3$ to the left of $\tan (θ)$ .

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

Step 4

Simplify $\tan (θ) \cdot \frac{3}{3}$ .

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Step 4.1

Divide $3$ by $3$ .

$\tan (θ) \cdot 1 = \frac{\sqrt{3}}{3}$

Step 4.2

Multiply $\tan (θ)$ by $1$ .

$\tan (θ) = \frac{\sqrt{3}}{3}$

Step 5

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

$θ = \arctan (\frac{\sqrt{3}}{3})$

Step 6

Simplify the right side.

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Step 6.1

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

$θ = \frac{π}{6}$

Step 7

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.

$θ = π + \frac{π}{6}$

Step 8

Simplify $π + \frac{π}{6}$ .

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Step 8.1

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

$θ = π \cdot \frac{6}{6} + \frac{π}{6}$

Step 8.2

Combine fractions.

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Step 8.2.1

Combine $π$ and $\frac{6}{6}$ .

$θ = \frac{π \cdot 6}{6} + \frac{π}{6}$

Step 8.2.2

Combine the numerators over the common denominator.

$θ = \frac{π \cdot 6 + π}{6}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Move $6$ to the left of $π$ .

$θ = \frac{6 \cdot π + π}{6}$

Step 8.3.2

Add $6 π$ and $π$ .

$θ = \frac{7 π}{6}$

Step 9

Find the period of $\tan (θ)$ .

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Step 9.1

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

$\frac{π}{| b |}$

Step 9.2

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

$\frac{π}{| 1 |}$

Step 9.3

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

$\frac{π}{1}$

Step 9.4

Divide $π$ by $1$ .

$π$

Step 10

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

$θ = \frac{π}{6} + π n, \frac{7 π}{6} + π n$ , for any integer $n$

Step 11

Consolidate the answers.

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