Precalculus Examples

$f (x) = \frac{\cos (x)}{1 - \sin (x)}$

Step 1

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

$1 - \sin (x) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $1$ from both sides of the equation.

$- \sin (x) = - 1$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2

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

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 1$ by $- 1$ .

$\sin (x) = 1$

Step 2.3

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

$x = \arcsin (1)$

Step 2.4

Simplify the right side.

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Step 2.4.1

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

$x = \frac{π}{2}$

Step 2.5

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.

$x = π - \frac{π}{2}$

Step 2.6

Simplify $π - \frac{π}{2}$ .

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Step 2.6.1

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

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

Step 2.6.2

Combine fractions.

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Step 2.6.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 2.6.3

Simplify the numerator.

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Step 2.6.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 2.6.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 2.7

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

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

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

Step 3

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

Set-Builder Notation:

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

Step 4