Trigonometry Examples

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

Step 1

Multiply both sides of the equation by $3$ .

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

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $3 (\frac{1}{3} \tan (x))$ .

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Step 2.1.1.1

Combine $\frac{1}{3}$ and $\tan (x)$ .

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

Step 2.1.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.2.1

Cancel the common factor.

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

Step 2.1.1.2.2

Rewrite the expression.

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 3

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

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

Step 4

Simplify the right side.

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Step 4.1

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

$x = \frac{π}{3}$

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.

$x = π + \frac{π}{3}$

Step 6

Simplify $π + \frac{π}{3}$ .

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Step 6.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 6.2

Combine fractions.

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Step 6.2.1

Combine $π$ and $\frac{3}{3}$ .

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

Step 6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 + π}{3}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π + π}{3}$

Step 6.3.2

Add $3 π$ and $π$ .

$x = \frac{4 π}{3}$

Step 7

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

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Step 7.1

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

$\frac{π}{| b |}$

Step 7.2

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

$\frac{π}{| 1 |}$

Step 7.3

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

$\frac{π}{1}$

Step 7.4

Divide $π$ by $1$ .

$π$

Step 8

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

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

Step 9

Consolidate the answers.

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