Trigonometry Examples

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

Step 1

Set the denominator in $\frac{3}{\tan (x) - \sqrt{3}}$ equal to $0$ to find where the expression is undefined.

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

Step 2

Solve for $x$ .

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Step 2.1

Add $\sqrt{3}$ to both sides of the equation.

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

Step 2.2

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

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$x = \frac{π}{3}$

Step 2.4

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.

$x = π + \frac{π}{3}$

Step 2.5

Simplify $π + \frac{π}{3}$ .

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Step 2.5.1

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 2.5.2

Combine fractions.

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Step 2.5.2.1

Combine $π$ and $\frac{3}{3}$ .

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

Step 2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 + π}{3}$

Step 2.5.3

Simplify the numerator.

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Step 2.5.3.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π + π}{3}$

Step 2.5.3.2

Add $3 π$ and $π$ .

$x = \frac{4 π}{3}$

Step 2.6

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

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Step 2.6.1

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

$\frac{π}{| b |}$

Step 2.6.2

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

$\frac{π}{| 1 |}$

Step 2.6.3

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

$\frac{π}{1}$

Step 2.6.4

Divide $π$ by $1$ .

$π$

Step 2.7

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 2.8

Consolidate the answers.

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

Step 3

Set the argument in $\tan (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x = \frac{π}{2} + π n$ , for any integer $n$

Step 4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = \frac{π}{3} + π n, x = \frac{π}{2} + π n}$ $n$ , for any integer $n$

Step 5