Calculus Examples

$x + \cot (\frac{x}{2})$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x + \cot (\frac{x}{2})$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\cot (\frac{x}{2})]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\cot (\frac{x}{2})]$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [\cot (\frac{x}{2})]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\cot (\frac{x}{2})]$ .

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Step 1.1.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cot (x)$ and $g (x) = \frac{x}{2}$ .

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Step 1.1.2.1.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$1 + \frac{d}{d u} [\cot (u)] \frac{d}{d x} [\frac{x}{2}]$

Step 1.1.2.1.2

The derivative of $\cot (u)$ with respect to $u$ is $- \csc^{2} (u)$ .

$1 - \csc^{2} (u) \frac{d}{d x} [\frac{x}{2}]$

Step 1.1.2.1.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$1 - \csc^{2} (\frac{x}{2}) \frac{d}{d x} [\frac{x}{2}]$

Step 1.1.2.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$1 - \csc^{2} (\frac{x}{2}) (\frac{1}{2} \frac{d}{d x} [x])$

Step 1.1.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 - \csc^{2} (\frac{x}{2}) (\frac{1}{2} \cdot 1)$

Step 1.1.2.4

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

$1 - \csc^{2} (\frac{x}{2}) \frac{1}{2}$

Step 1.1.2.5

Combine $\frac{1}{2}$ and $\csc^{2} (\frac{x}{2})$ .

$1 - \frac{\csc^{2} (\frac{x}{2})}{2}$

Step 1.1.3

Reorder terms.

$f' (x) = - \frac{\csc^{2} (\frac{x}{2})}{2} + 1$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{\csc^{2} (\frac{x}{2})}{2} + 1$ .

$- \frac{\csc^{2} (\frac{x}{2})}{2} + 1$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{\csc^{2} (\frac{x}{2})}{2} + 1 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{\csc^{2} (\frac{x}{2})}{2} + 1 = 0$

Step 2.2

Subtract $1$ from both sides of the equation.

$- \frac{\csc^{2} (\frac{x}{2})}{2} = - 1$

Step 2.3

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{\csc^{2} (\frac{x}{2})}{2}) = - 2 \cdot - 1$

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify $- 2 (- \frac{\csc^{2} (\frac{x}{2})}{2})$ .

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Step 2.4.1.1.1

Cancel the common factor of $2$ .

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Step 2.4.1.1.1.1

Move the leading negative in $- \frac{\csc^{2} (\frac{x}{2})}{2}$ into the numerator.

$- 2 \frac{- \csc^{2} (\frac{x}{2})}{2} = - 2 \cdot - 1$

Step 2.4.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- \csc^{2} (\frac{x}{2})}{2} = - 2 \cdot - 1$

Step 2.4.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- \csc^{2} (\frac{x}{2})}{2} = - 2 \cdot - 1$

Step 2.4.1.1.1.4

Rewrite the expression.

$- - \csc^{2} (\frac{x}{2}) = - 2 \cdot - 1$

Step 2.4.1.1.2

Multiply.

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Step 2.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 \csc^{2} (\frac{x}{2}) = - 2 \cdot - 1$

Step 2.4.1.1.2.2

Multiply $\csc^{2} (\frac{x}{2})$ by $1$ .

$\csc^{2} (\frac{x}{2}) = - 2 \cdot - 1$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $- 2$ by $- 1$ .

$\csc^{2} (\frac{x}{2}) = 2$

Step 2.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\csc (\frac{x}{2}) = \pm \sqrt{2}$

Step 2.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.6.1

First, use the positive value of the $\pm$ to find the first solution.

$\csc (\frac{x}{2}) = \sqrt{2}$

Step 2.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$\csc (\frac{x}{2}) = - \sqrt{2}$

Step 2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$\csc (\frac{x}{2}) = \sqrt{2}, - \sqrt{2}$

Step 2.7

Set up each of the solutions to solve for $x$ .

$\csc (\frac{x}{2}) = \sqrt{2}$

$\csc (\frac{x}{2}) = - \sqrt{2}$

Step 2.8

Solve for $x$ in $\csc (\frac{x}{2}) = \sqrt{2}$ .

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Step 2.8.1

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

$\frac{x}{2} = arccsc (\sqrt{2})$

Step 2.8.2

Simplify the right side.

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Step 2.8.2.1

The exact value of $arccsc (\sqrt{2})$ is $\frac{π}{4}$ .

$\frac{x}{2} = \frac{π}{4}$

Step 2.8.3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{π}{4}$

Step 2.8.4

Simplify both sides of the equation.

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Step 2.8.4.1

Simplify the left side.

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Step 2.8.4.1.1

Cancel the common factor of $2$ .

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Step 2.8.4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{π}{4}$

Step 2.8.4.1.1.2

Rewrite the expression.

$x = 2 \frac{π}{4}$

Step 2.8.4.2

Simplify the right side.

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Step 2.8.4.2.1

Cancel the common factor of $2$ .

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Step 2.8.4.2.1.1

Factor $2$ out of $4$ .

$x = 2 \frac{π}{2 (2)}$

Step 2.8.4.2.1.2

Cancel the common factor.

$x = 2 \frac{π}{2 \cdot 2}$

Step 2.8.4.2.1.3

Rewrite the expression.

$x = \frac{π}{2}$

Step 2.8.5

The cosecant 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.

$\frac{x}{2} = π - \frac{π}{4}$

Step 2.8.6

Solve for $x$ .

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Step 2.8.6.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π - \frac{π}{4})$

Step 2.8.6.2

Simplify both sides of the equation.

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Step 2.8.6.2.1

Simplify the left side.

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Step 2.8.6.2.1.1

Cancel the common factor of $2$ .

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Step 2.8.6.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π - \frac{π}{4})$

Step 2.8.6.2.1.1.2

Rewrite the expression.

$x = 2 (π - \frac{π}{4})$

Step 2.8.6.2.2

Simplify the right side.

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Step 2.8.6.2.2.1

Simplify $2 (π - \frac{π}{4})$ .

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Step 2.8.6.2.2.1.1

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

$x = 2 (π \cdot \frac{4}{4} - \frac{π}{4})$

Step 2.8.6.2.2.1.2

Simplify terms.

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Step 2.8.6.2.2.1.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = 2 (\frac{π \cdot 4}{4} - \frac{π}{4})$

Step 2.8.6.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 2 \frac{π \cdot 4 - π}{4}$

Step 2.8.6.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 2.8.6.2.2.1.2.3.1

Factor $2$ out of $4$ .

$x = 2 \frac{π \cdot 4 - π}{2 (2)}$

Step 2.8.6.2.2.1.2.3.2

Cancel the common factor.

$x = 2 \frac{π \cdot 4 - π}{2 \cdot 2}$

Step 2.8.6.2.2.1.2.3.3

Rewrite the expression.

$x = \frac{π \cdot 4 - π}{2}$

Step 2.8.6.2.2.1.3

Simplify the numerator.

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Step 2.8.6.2.2.1.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π - π}{2}$

Step 2.8.6.2.2.1.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 2.8.7

Find the period of $\csc (\frac{x}{2})$ .

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Step 2.8.7.1

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

$\frac{2 π}{| b |}$

Step 2.8.7.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 2.8.7.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 2.8.7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 2.8.7.5

Multiply $2$ by $2$ .

$4 π$

Step 2.8.8

The period of the $\csc (\frac{x}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = \frac{π}{2} + 4 π n, \frac{3 π}{2} + 4 π n$ , for any integer $n$

Step 2.9

Solve for $x$ in $\csc (\frac{x}{2}) = - \sqrt{2}$ .

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Step 2.9.1

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

$\frac{x}{2} = arccsc (- \sqrt{2})$

Step 2.9.2

Simplify the right side.

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Step 2.9.2.1

The exact value of $arccsc (- \sqrt{2})$ is $- \frac{π}{4}$ .

$\frac{x}{2} = - \frac{π}{4}$

Step 2.9.3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (- \frac{π}{4})$

Step 2.9.4

Simplify both sides of the equation.

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Step 2.9.4.1

Simplify the left side.

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Step 2.9.4.1.1

Cancel the common factor of $2$ .

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Step 2.9.4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (- \frac{π}{4})$

Step 2.9.4.1.1.2

Rewrite the expression.

$x = 2 (- \frac{π}{4})$

Step 2.9.4.2

Simplify the right side.

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Step 2.9.4.2.1

Simplify $2 (- \frac{π}{4})$ .

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Step 2.9.4.2.1.1

Cancel the common factor of $2$ .

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Step 2.9.4.2.1.1.1

Move the leading negative in $- \frac{π}{4}$ into the numerator.

$x = 2 \frac{- π}{4}$

Step 2.9.4.2.1.1.2

Factor $2$ out of $4$ .

$x = 2 \frac{- π}{2 (2)}$

Step 2.9.4.2.1.1.3

Cancel the common factor.

$x = 2 \frac{- π}{2 \cdot 2}$

Step 2.9.4.2.1.1.4

Rewrite the expression.

$x = \frac{- π}{2}$

Step 2.9.4.2.1.2

Move the negative in front of the fraction.

$x = - \frac{π}{2}$

Step 2.9.5

The cosecant 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.

$\frac{x}{2} = 2 π + \frac{π}{4} + π$

Step 2.9.6

Simplify the expression to find the second solution.

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Step 2.9.6.1

Subtract $2 π$ from $2 π + \frac{π}{4} + π$ .

$\frac{x}{2} = 2 π + \frac{π}{4} + π - 2 π$

Step 2.9.6.2

The resulting angle of $\frac{5 π}{4}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{4} + π$ .

$\frac{x}{2} = \frac{5 π}{4}$

Step 2.9.6.3

Solve for $x$ .

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Step 2.9.6.3.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{5 π}{4}$

Step 2.9.6.3.2

Simplify both sides of the equation.

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Step 2.9.6.3.2.1

Simplify the left side.

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Step 2.9.6.3.2.1.1

Cancel the common factor of $2$ .

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Step 2.9.6.3.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{5 π}{4}$

Step 2.9.6.3.2.1.1.2

Rewrite the expression.

$x = 2 \frac{5 π}{4}$

Step 2.9.6.3.2.2

Simplify the right side.

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Step 2.9.6.3.2.2.1

Cancel the common factor of $2$ .

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Step 2.9.6.3.2.2.1.1

Factor $2$ out of $4$ .

$x = 2 \frac{5 π}{2 (2)}$

Step 2.9.6.3.2.2.1.2

Cancel the common factor.

$x = 2 \frac{5 π}{2 \cdot 2}$

Step 2.9.6.3.2.2.1.3

Rewrite the expression.

$x = \frac{5 π}{2}$

Step 2.9.7

Find the period of $\csc (\frac{x}{2})$ .

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Step 2.9.7.1

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

$\frac{2 π}{| b |}$

Step 2.9.7.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 2.9.7.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 2.9.7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 2.9.7.5

Multiply $2$ by $2$ .

$4 π$

Step 2.9.8

Add $4 π$ to every negative angle to get positive angles.

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Step 2.9.8.1

Add $4 π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + 4 π$

Step 2.9.8.2

To write $4 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 2.9.8.3

Combine fractions.

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Step 2.9.8.3.1

Combine $4 π$ and $\frac{2}{2}$ .

$\frac{4 π \cdot 2}{2} - \frac{π}{2}$

Step 2.9.8.3.2

Combine the numerators over the common denominator.

$\frac{4 π \cdot 2 - π}{2}$

Step 2.9.8.4

Simplify the numerator.

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Step 2.9.8.4.1

Multiply $2$ by $4$ .

$\frac{8 π - π}{2}$

Step 2.9.8.4.2

Subtract $π$ from $8 π$ .

$\frac{7 π}{2}$

Step 2.9.8.5

List the new angles.

$x = \frac{7 π}{2}$

Step 2.9.9

The period of the $\csc (\frac{x}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = \frac{5 π}{2} + 4 π n, \frac{7 π}{2} + 4 π n$ , for any integer $n$

Step 2.10

List all of the solutions.

$x = \frac{π}{2} + 4 π n, \frac{3 π}{2} + 4 π n, \frac{5 π}{2} + 4 π n, \frac{7 π}{2} + 4 π n$ , for any integer $n$

Step 2.11

Consolidate the answers.

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the argument in $\csc (\frac{x}{2})$ equal to $π n$ to find where the expression is undefined.

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π n)$

Step 3.2.2

Simplify both sides of the equation.

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π n)$

Step 3.2.2.1.1.2

Rewrite the expression.

$x = 2 (π n)$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Remove parentheses.

$x = 2 π n$

Step 3.3

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 $0$ .

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

Step 4

Evaluate $x + \cot (\frac{x}{2})$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{π}{2}$ .

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Step 4.1.1

Substitute $\frac{π}{2}$ for $x$ .

$(\frac{π}{2}) + \cot (\frac{\frac{π}{2}}{2})$

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{π}{2} + \cot (\frac{π}{2} \cdot \frac{1}{2})$

Step 4.1.2.2

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

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Step 4.1.2.2.1

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

$\frac{π}{2} + \cot (\frac{π}{2 \cdot 2})$

Step 4.1.2.2.2

Multiply $2$ by $2$ .

$\frac{π}{2} + \cot (\frac{π}{4})$

Step 4.1.2.3

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$\frac{π}{2} + 1$

Step 4.2

Evaluate at $x = \frac{3 π}{2}$ .

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Step 4.2.1

Substitute $\frac{3 π}{2}$ for $x$ .

$(\frac{3 π}{2}) + \cot (\frac{\frac{3 π}{2}}{2})$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 π}{2} + \cot (\frac{3 π}{2} \cdot \frac{1}{2})$

Step 4.2.2.2

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

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Step 4.2.2.2.1

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

$\frac{3 π}{2} + \cot (\frac{3 π}{2 \cdot 2})$

Step 4.2.2.2.2

Multiply $2$ by $2$ .

$\frac{3 π}{2} + \cot (\frac{3 π}{4})$

Step 4.2.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cotangent is negative in the second quadrant.

$\frac{3 π}{2} - \cot (\frac{π}{4})$

Step 4.2.2.4

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$\frac{3 π}{2} - 1 \cdot 1$

Step 4.2.2.5

Multiply $- 1$ by $1$ .

$\frac{3 π}{2} - 1$

Step 4.3

Evaluate at $x = \frac{5 π}{2}$ .

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Step 4.3.1

Substitute $\frac{5 π}{2}$ for $x$ .

$(\frac{5 π}{2}) + \cot (\frac{\frac{5 π}{2}}{2})$

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 π}{2} + \cot (\frac{5 π}{2} \cdot \frac{1}{2})$

Step 4.3.2.2

Multiply $\frac{5 π}{2} \cdot \frac{1}{2}$ .

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Step 4.3.2.2.1

Multiply $\frac{5 π}{2}$ by $\frac{1}{2}$ .

$\frac{5 π}{2} + \cot (\frac{5 π}{2 \cdot 2})$

Step 4.3.2.2.2

Multiply $2$ by $2$ .

$\frac{5 π}{2} + \cot (\frac{5 π}{4})$

Step 4.3.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{5 π}{2} + \cot (\frac{π}{4})$

Step 4.3.2.4

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$\frac{5 π}{2} + 1$

Step 4.4

Evaluate at $x = \frac{7 π}{2}$ .

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Step 4.4.1

Substitute $\frac{7 π}{2}$ for $x$ .

$(\frac{7 π}{2}) + \cot (\frac{\frac{7 π}{2}}{2})$

Step 4.4.2

Simplify each term.

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Step 4.4.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{7 π}{2} + \cot (\frac{7 π}{2} \cdot \frac{1}{2})$

Step 4.4.2.2

Multiply $\frac{7 π}{2} \cdot \frac{1}{2}$ .

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Step 4.4.2.2.1

Multiply $\frac{7 π}{2}$ by $\frac{1}{2}$ .

$\frac{7 π}{2} + \cot (\frac{7 π}{2 \cdot 2})$

Step 4.4.2.2.2

Multiply $2$ by $2$ .

$\frac{7 π}{2} + \cot (\frac{7 π}{4})$

Step 4.4.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cotangent is negative in the fourth quadrant.

$\frac{7 π}{2} - \cot (\frac{π}{4})$

Step 4.4.2.4

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$\frac{7 π}{2} - 1 \cdot 1$

Step 4.4.2.5

Multiply $- 1$ by $1$ .

$\frac{7 π}{2} - 1$

Step 4.5

Evaluate at $x = \frac{9 π}{2}$ .

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Step 4.5.1

Substitute $\frac{9 π}{2}$ for $x$ .

$(\frac{9 π}{2}) + \cot (\frac{\frac{9 π}{2}}{2})$

Step 4.5.2

Simplify each term.

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Step 4.5.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{9 π}{2} + \cot (\frac{9 π}{2} \cdot \frac{1}{2})$

Step 4.5.2.2

Multiply $\frac{9 π}{2} \cdot \frac{1}{2}$ .

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Step 4.5.2.2.1

Multiply $\frac{9 π}{2}$ by $\frac{1}{2}$ .

$\frac{9 π}{2} + \cot (\frac{9 π}{2 \cdot 2})$

Step 4.5.2.2.2

Multiply $2$ by $2$ .

$\frac{9 π}{2} + \cot (\frac{9 π}{4})$

Step 4.5.2.3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{9 π}{2} + \cot (\frac{π}{4})$

Step 4.5.2.4

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$\frac{9 π}{2} + 1$

Step 4.6

List all of the points.

$(\frac{π}{2}, \frac{π}{2} + 1), (\frac{3 π}{2}, \frac{3 π}{2} - 1), (\frac{5 π}{2}, \frac{5 π}{2} + 1), (\frac{7 π}{2}, \frac{7 π}{2} - 1), (\frac{9 π}{2}, \frac{9 π}{2} + 1)$

Step 5