Calculus Examples

$f (x) = \frac{x}{\cos (x)}$

Step 1

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

$\cos (x) = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

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

$x = \arccos (0)$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 2.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 2.4

Simplify $2 π - \frac{π}{2}$ .

Tap for more steps...

Step 2.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 2.4.2

Combine fractions.

Tap for more steps...

Step 2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 2.4.3

Simplify the numerator.

Tap for more steps...

Step 2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 2.5

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

Tap for more steps...

Step 2.5.1

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

$\frac{2 π}{| b |}$

Step 2.5.2

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

$\frac{2 π}{| 1 |}$

Step 2.5.3

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

$\frac{2 π}{1}$

Step 2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.6

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

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 2.7

Consolidate the answers.

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

Step 3

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

Set-Builder Notation:

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

Step 4