Trigonometry Examples

$\cot (x)$

Step 1

For any $y = \cot (x)$ , vertical asymptotes occur at $x = n π$ , where $n$ is an integer. Use the basic period for $y = \cot (x)$ , $(0, π)$ , to find the vertical asymptotes for $y = \cot (x)$ . Set the inside of the cotangent function, $b x + c$ , for $y = a \cot (b x + c) + d$ equal to $0$ to find where the vertical asymptote occurs for $y = \cot (x)$ .

$x = 0$

Step 2

Set the inside of the cotangent function $x$ equal to $π$ .

$x = π$

Step 3

The basic period for $y = \cot (x)$ will occur at $(0, π)$ , where $0$ and $π$ are vertical asymptotes.

$(0, π)$

Step 4

Find the period $\frac{π}{| b |}$ to find where the vertical asymptotes exist.

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Step 4.1

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

$\frac{π}{1}$

Step 4.2

Divide $π$ by $1$ .

$π$

Step 5

The vertical asymptotes for $y = \cot (x)$ occur at $0$ , $π$ , and every $π n$ , where $n$ is an integer.

$π n$

Step 6

There are only vertical asymptotes for tangent and cotangent functions.

Vertical Asymptotes: $x = π n$ for any integer $n$

No Horizontal Asymptotes

No Oblique Asymptotes

Step 7