Trigonometry Examples

$f (x) = | \cot (x) - 2 |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = | \cot (x) - 2 |$ is $(0.4636476 + π n, | \cot (0.4636476 + π n) - 2 |)$ .

Tap for more steps...

Step 1.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $\cot (x) - 2$ equal to $0$ . In this case, $\cot (x) - 2 = 0$ .

$\cot (x) - 2 = 0$

Step 1.2

Solve the equation $\cot (x) - 2 = 0$ to find the $x$ coordinate for the absolute value vertex.

Tap for more steps...

Step 1.2.1

Add $2$ to both sides of the equation.

$\cot (x) = 2$

Step 1.2.2

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

$x = arccot (2)$

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Evaluate $arccot (2)$ .

$x = 0.4636476$

Step 1.2.4

The cotangent 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.

$x = (3.14159265) + 0.4636476$

Step 1.2.5

Solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Remove parentheses.

$x = 3.14159265 + 0.4636476$

Step 1.2.5.2

Remove parentheses.

$x = (3.14159265) + 0.4636476$

Step 1.2.5.3

Add $3.14159265$ and $0.4636476$ .

$x = 3.60524026$

Step 1.2.6

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

Tap for more steps...

Step 1.2.6.1

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

$\frac{π}{| b |}$

Step 1.2.6.2

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

$\frac{π}{| 1 |}$

Step 1.2.6.3

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

$\frac{π}{1}$

Step 1.2.6.4

Divide $π$ by $1$ .

$π$

Step 1.2.7

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

$x = 0.4636476 + π n, 3.60524026 + π n$ , for any integer $n$

Step 1.2.8

Consolidate $0.4636476 + π n$ and $3.60524026 + π n$ to $0.4636476 + π n$ .

$x = 0.4636476 + π n$ , for any integer $n$

Step 1.3

Replace the variable $x$ with $0.4636476 + π n$ in the expression.

$y = | \cot (0.4636476 + π n) - 2 |$

Step 1.4

The absolute value vertex is $(0.4636476 + π n, | \cot (0.4636476 + π n) - 2 |)$ .

$(0.4636476 + π n, | \cot (0.4636476 + π n) - 2 |)$

Step 2

Find the domain for $f (x) = | \cot (x) - 2 |$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the absolute value function.

Tap for more steps...

Step 2.1

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

$x = π n$ , for any integer $n$

Step 2.2

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

Set-Builder Notation:

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

Set-Builder Notation:

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

Step 3

The absolute value can be graphed using the points around the vertex $(, | \cot (0.4636476 + π n) - 2 |),$

$\begin{array}{c} x & y \\ N a N \end{array}$

Step 4