Trigonometry Examples

$y = \cot (\frac{4}{3} x)$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the argument in $\cot (\frac{4}{3} x)$ equal to $π n$ to find where the expression is undefined.

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

Step 1.2

Solve for $x$ .

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Step 1.2.1

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} (\frac{4}{3} x) = \frac{3}{4} (π n)$

Step 1.2.2

Simplify both sides of the equation.

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Step 1.2.2.1

Simplify the left side.

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Step 1.2.2.1.1

Simplify $\frac{3}{4} (\frac{4}{3} x)$ .

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Step 1.2.2.1.1.1

Combine $\frac{4}{3}$ and $x$ .

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} (π n)$

Step 1.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.2.1.1.2.1

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} (π n)$

Step 1.2.2.1.1.2.2

Rewrite the expression.

$\frac{1}{4} (4 x) = \frac{3}{4} (π n)$

Step 1.2.2.1.1.3

Cancel the common factor of $4$ .

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Step 1.2.2.1.1.3.1

Factor $4$ out of $4 x$ .

$\frac{1}{4} (4 (x)) = \frac{3}{4} (π n)$

Step 1.2.2.1.1.3.2

Cancel the common factor.

$\frac{1}{4} (4 x) = \frac{3}{4} (π n)$

Step 1.2.2.1.1.3.3

Rewrite the expression.

$x = \frac{3}{4} (π n)$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Simplify $\frac{3}{4} (π n)$ .

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Step 1.2.2.2.1.1

Multiply $\frac{3}{4} (π n)$ .

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Step 1.2.2.2.1.1.1

Combine $π$ and $\frac{3}{4}$ .

$x = \frac{π \cdot 3}{4} n$

Step 1.2.2.2.1.1.2

Combine $\frac{π \cdot 3}{4}$ and $n$ .

$x = \frac{π \cdot 3 n}{4}$

Step 1.2.2.2.1.2

Move $3$ to the left of $π$ .

$x = \frac{3 π n}{4}$

Step 1.3

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

Set-Builder Notation:

${x | x \neq \frac{3 π n}{4}}$ , for any integer $n$

Set-Builder Notation:

${x | x \neq \frac{3 π n}{4}}$ , for any integer $n$

Step 2

Since the domain is not all real numbers, $\cot (\frac{4}{3} x)$ is not continuous over all real numbers.

Not continuous

Step 3