Calculus Examples

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

Step 1

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

$\sin (2 x) = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$2 x = \arcsin (0)$

Step 2.2

Simplify the right side.

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Step 2.2.1

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

$2 x = 0$

Step 2.3

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 2.4

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.

$2 x = π - 0$

Step 2.5

Solve for $x$ .

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Step 2.5.1

Simplify.

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Step 2.5.1.1

Multiply $- 1$ by $0$ .

$2 x = π + 0$

Step 2.5.1.2

Add $π$ and $0$ .

$2 x = π$

Step 2.5.2

Divide each term in $2 x = π$ by $2$ and simplify.

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Step 2.5.2.1

Divide each term in $2 x = π$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Step 2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2}$

Step 2.6

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

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Step 2.6.1

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

$\frac{2 π}{| b |}$

Step 2.6.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 2.6.3

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

$\frac{2 π}{2}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 2.6.4.2

Divide $π$ by $1$ .

$π$

Step 2.7

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

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

Step 2.8

Consolidate the answers.

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

Step 3

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

Set-Builder Notation:

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

Step 4