Precalculus Examples

$\frac{\sin^{2} (y)}{1 - \cos (y)}$

Step 1

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

$1 - \cos (y) = 0$

Step 2

Solve for $y$ .

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Step 2.1

Subtract $1$ from both sides of the equation.

$- \cos (y) = - 1$

Step 2.2

Divide each term in $- \cos (y) = - 1$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- \cos (y) = - 1$ by $- 1$ .

$\frac{- \cos (y)}{- 1} = \frac{- 1}{- 1}$

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.

$\frac{\cos (y)}{1} = \frac{- 1}{- 1}$

Step 2.2.2.2

Divide $\cos (y)$ by $1$ .

$\cos (y) = \frac{- 1}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 1$ by $- 1$ .

$\cos (y) = 1$

Step 2.3

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

$y = \arccos (1)$

Step 2.4

Simplify the right side.

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Step 2.4.1

The exact value of $\arccos (1)$ is $0$ .

$y = 0$

Step 2.5

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.

$y = 2 π - 0$

Step 2.6

Subtract $0$ from $2 π$ .

$y = 2 π$

Step 2.7

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

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

$y = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 2.9

Consolidate the answers.

$y = 2 π n$ , for any integer $n$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${y | y = 2 π n}$ $n$ , for any integer $n$

Step 4