Calculus Examples

$\frac{\sin (2 x)}{1 + \sin (2 x)}$

Step 1

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

$1 + \sin (2 x) = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Subtract $1$ from both sides of the equation.

$\sin (2 x) = - 1$

Step 2.2

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

$2 x = \arcsin (- 1)$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

The exact value of $\arcsin (- 1)$ is $- \frac{π}{2}$ .

$2 x = - \frac{π}{2}$

Step 2.4

Divide each term in $2 x = - \frac{π}{2}$ by $2$ and simplify.

Tap for more steps...

Step 2.4.1

Divide each term in $2 x = - \frac{π}{2}$ by $2$ .

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

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{2}}{2}$

Step 2.4.3

Simplify the right side.

Tap for more steps...

Step 2.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{2} \cdot \frac{1}{2}$

Step 2.4.3.2

Multiply $- \frac{π}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 2.4.3.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{2}$ .

$x = - \frac{π}{2 \cdot 2}$

Step 2.4.3.2.2

Multiply $2$ by $2$ .

$x = - \frac{π}{4}$

Step 2.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$2 x = 2 π + \frac{π}{2} + π$

Step 2.6

Simplify the expression to find the second solution.

Tap for more steps...

Step 2.6.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$2 x = 2 π + \frac{π}{2} + π - 2 π$

Step 2.6.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$2 x = \frac{3 π}{2}$

Step 2.6.3

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ and simplify.

Tap for more steps...

Step 2.6.3.1

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 2.6.3.2

Simplify the left side.

Tap for more steps...

Step 2.6.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 2.6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 2.6.3.3

Simplify the right side.

Tap for more steps...

Step 2.6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 2.6.3.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 2.6.3.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{2}$ .

$x = \frac{3 π}{2 \cdot 2}$

Step 2.6.3.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 2.7

Find the period of $\sin (2 x)$ .

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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 2.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 2.7.4.2

Divide $π$ by $1$ .

$π$

Step 2.8

Add $π$ to every negative angle to get positive angles.

Tap for more steps...

Step 2.8.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 2.8.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 2.8.3

Combine fractions.

Tap for more steps...

Step 2.8.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 2.8.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 2.8.4

Simplify the numerator.

Tap for more steps...

Step 2.8.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 2.8.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 2.8.5

List the new angles.

$x = \frac{3 π}{4}$

Step 2.9

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

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 2.10

Consolidate the answers.

$x = \frac{3 π}{4} + π n$ , for any integer $n$

Step 3

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

Set-Builder Notation:

${x | x \neq \frac{3 π}{4} + π n}$ , 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: ${x | x \neq \frac{3 π}{4} + π n}$ , for any integer $n$

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

Step 6