Find the Domain and Range y=1/(sin(x))

$y = \frac{1}{\sin (x)}$

Step 1

Set the denominator in $\frac{1}{\sin (x)}$ equal to $0$ to find where the expression is undefined.

$\sin (x) = 0$

Step 2

Solve for $x$ .

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

$x = \arcsin (0)$

Simplify the right side.

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = π - 0$

Subtract $0$ from $π$ .

$x = π$

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

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2 π n, π + 2 π n$ , for any integer $n$

Consolidate the answers.

$x = π n$ , for any integer $n$

Step 3

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

Set-Builder Notation:

${x | x \neq π n}$ , 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:

$(- \infty, - 1] \cup [1, \infty)$

Set-Builder Notation:

${y | y \leq - 1, y \geq 1}$

Step 5

Determine the domain and range.

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

Range: $(- \infty, - 1] \cup [1, \infty), {y | y \leq - 1, y \geq 1}$

Step 6

Find the Domain and Range y=1/(sin(x))

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