Calculus Examples

$\csc (x)$

Step 1

Find the derivative.

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Step 1.1

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$- \csc (x) \cot (x)$

Step 1.2

Reorder the factors of $- \csc (x) \cot (x)$ .

$- \cot (x) \csc (x)$

Step 2

Set the derivative equal to $0$ then solve the equation $- \cot (x) \csc (x) = 0$ .

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Step 2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cot (x) = 0$

$\csc (x) = 0$

Step 2.2

Set $\cot (x)$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $\cot (x)$ equal to $0$ .

$\cot (x) = 0$

Step 2.2.2

Solve $\cot (x) = 0$ for $x$ .

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Step 2.2.2.1

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

$x = arccot (0)$

Step 2.2.2.2

Simplify the right side.

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Step 2.2.2.2.1

The exact value of $arccot (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 2.2.2.3

The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{2}$

Step 2.2.2.4

Simplify $π + \frac{π}{2}$ .

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Step 2.2.2.4.1

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 2.2.2.4.2

Combine fractions.

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Step 2.2.2.4.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 2.2.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 + π}{2}$

Step 2.2.2.4.3

Simplify the numerator.

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Step 2.2.2.4.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π + π}{2}$

Step 2.2.2.4.3.2

Add $2 π$ and $π$ .

$x = \frac{3 π}{2}$

Step 2.2.2.5

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

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Step 2.2.2.5.1

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

$\frac{π}{| b |}$

Step 2.2.2.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 2.2.2.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 2.2.2.5.4

Divide $π$ by $1$ .

$π$

Step 2.2.2.6

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

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

Step 2.3

Set $\csc (x)$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $\csc (x)$ equal to $0$ .

$\csc (x) = 0$

Step 2.3.2

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 2.4

The final solution is all the values that make $- \cot (x) \csc (x) = 0$ true.

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

Step 2.5

Consolidate the answers.

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

Step 3

Solve the original function $f (x) = \csc (x)$ at $x = \frac{π}{2}$ .

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Step 3.1

Replace the variable $x$ with $\frac{π}{2}$ in the expression.

$f (\frac{π}{2}) = \csc (\frac{π}{2})$

Step 3.2

Simplify the result.

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Step 3.2.1

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

$f (\frac{π}{2}) = 1$

Step 3.2.2

The final answer is $1$ .

$1$

Step 4

Solve the original function $f (x) = \csc (x)$ at $x = \frac{π}{2} + π$ .

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Step 4.1

Replace the variable $x$ with $\frac{π}{2} + π$ in the expression.

$f (\frac{π}{2} + π) = \csc (\frac{π}{2} + π)$

Step 4.2

Simplify the result.

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Step 4.2.1

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

$f (\frac{π}{2} + π) = \csc (\frac{π}{2} + π \cdot \frac{2}{2})$

Step 4.2.2

Combine fractions.

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Step 4.2.2.1

Combine $π$ and $\frac{2}{2}$ .

$f (\frac{π}{2} + π) = \csc (\frac{π}{2} + \frac{π \cdot 2}{2})$

Step 4.2.2.2

Combine the numerators over the common denominator.

$f (\frac{π}{2} + π) = \csc (\frac{π + π \cdot 2}{2})$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Move $2$ to the left of $π$ .

$f (\frac{π}{2} + π) = \csc (\frac{π + 2 \cdot π}{2})$

Step 4.2.3.2

Add $π$ and $2 π$ .

$f (\frac{π}{2} + π) = \csc (\frac{3 π}{2})$

Step 4.2.4

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

$f (\frac{π}{2} + π) = - \csc (\frac{π}{2})$

Step 4.2.5

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

$f (\frac{π}{2} + π) = - 1 \cdot 1$

Step 4.2.6

Multiply $- 1$ by $1$ .

$f (\frac{π}{2} + π) = - 1$

Step 4.2.7

The final answer is $- 1$ .

$- 1$

Step 5

The horizontal tangent line on function $f (x) = \csc (x)$ is $y = 1, y = - 1$ .

$y = 1, y = - 1$

Step 6