Trigonometry Examples

$y = \frac{1}{2 - \csc (x)}$

Step 1

Set the denominator in $\frac{1}{2 - \csc (x)}$ equal to $0$ to find where the expression is undefined.

$2 - \csc (x) = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Subtract $2$ from both sides of the equation.

$- \csc (x) = - 2$

Step 2.2

Divide each term in $- \csc (x) = - 2$ by $- 1$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $- \csc (x) = - 2$ by $- 1$ .

$\frac{- \csc (x)}{- 1} = \frac{- 2}{- 1}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{\csc (x)}{1} = \frac{- 2}{- 1}$

Step 2.2.2.2

Divide $\csc (x)$ by $1$ .

$\csc (x) = \frac{- 2}{- 1}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Divide $- 2$ by $- 1$ .

$\csc (x) = 2$

Step 2.3

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

$x = arccsc (2)$

Step 2.4

Simplify the right side.

Tap for more steps...

Step 2.4.1

The exact value of $arccsc (2)$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 2.5

The cosecant 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.

$x = π - \frac{π}{6}$

Step 2.6

Simplify $π - \frac{π}{6}$ .

Tap for more steps...

Step 2.6.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 2.6.2

Combine fractions.

Tap for more steps...

Step 2.6.2.1

Combine $π$ and $\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 2.6.3

Simplify the numerator.

Tap for more steps...

Step 2.6.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 2.6.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

Step 2.7

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

Tap for more steps...

Step 2.7.1

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

$\frac{2 π}{| b |}$

Step 2.7.2

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

$\frac{2 π}{| 1 |}$

Step 2.7.3

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

$\frac{2 π}{1}$

Step 2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.8

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

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 3

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

$x = π n$ , for any integer $n$

Step 4

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

Set-Builder Notation:

${x | x \neq \frac{π}{6} + 2 π n, x \neq \frac{5 π}{6} + 2 π n, x \neq π n}$ , for any integer $n$

Step 5