Trigonometry Examples

$f (x) = 2 \csc (\frac{x}{2})$

Step 1

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

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

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π n)$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π n)$

Step 2.2.1.1.2

Rewrite the expression.

$x = 2 (π n)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Remove parentheses.

$x = 2 π n$

Step 3

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

Set-Builder Notation:

${x | x \neq 2 π n}$ , for any integer $n$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, - 2] \cup [2, \infty)$

Set-Builder Notation:

${y | y \leq - 2, y \geq 2}$

Step 5

Determine the domain and range.

Domain: ${x | x \neq 2 π n}$ , for any integer $n$

Range: $(- \infty, - 2] \cup [2, \infty), {y | y \leq - 2, y \geq 2}$

Step 6