Trigonometry Examples

$\tan (3 x)$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the argument in $\tan (3 x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

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

Step 1.2

Divide each term in $3 x = \frac{π}{2} + π n$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = \frac{π}{2} + π n$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{π}{2}}{3} + \frac{π n}{3}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{π}{2}}{3} + \frac{π n}{3}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{3} + \frac{π n}{3}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{3} + \frac{π n}{3}$

Step 1.2.3.1.2

Multiply $\frac{π}{2} \cdot \frac{1}{3}$ .

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Step 1.2.3.1.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{3}$ .

$x = \frac{π}{2 \cdot 3} + \frac{π n}{3}$

Step 1.2.3.1.2.2

Multiply $2$ by $3$ .

$x = \frac{π}{6} + \frac{π n}{3}$

Step 1.3

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

${x | x \neq \frac{π}{6} + \frac{π n}{3}}$ , for any integer $n$

Set-Builder Notation:

${x | x \neq \frac{π}{6} + \frac{π n}{3}}$ , for any integer $n$

Step 2

Since the domain is not all real numbers, $\tan (3 x)$ is not continuous over all real numbers.

Not continuous

Step 3