Precalculus Examples

$y = \cot (x)$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \cot (x)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\cot (x) = 0$ .

$\cot (x) = 0$

Step 1.2.2

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

$x = arccot (0)$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

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

$x = \frac{π}{2}$

Step 1.2.4

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 1.2.5

Simplify $π + \frac{π}{2}$ .

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Step 1.2.5.1

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 1.2.5.2

Combine fractions.

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Step 1.2.5.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 + π}{2}$

Step 1.2.5.3

Simplify the numerator.

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Step 1.2.5.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π + π}{2}$

Step 1.2.5.3.2

Add $2 π$ and $π$ .

$x = \frac{3 π}{2}$

Step 1.2.6

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

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Step 1.2.6.1

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

$\frac{π}{| b |}$

Step 1.2.6.2

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

$\frac{π}{| 1 |}$

Step 1.2.6.3

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

$\frac{π}{1}$

Step 1.2.6.4

Divide $π$ by $1$ .

$π$

Step 1.2.7

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 1.2.8

Consolidate the answers.

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

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{π}{2} + π n, 0)$ , for any integer $n$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \cot (0)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \cot (0)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $\cot (0)$ .

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Step 2.2.2.1.1

Rewrite $\cot (0)$ in terms of sines and cosines.

$y = \frac{\cos (0)}{\sin (0)}$

Step 2.2.2.1.2

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

$y = \frac{\cos (0)}{0}$

Step 2.2.2.2

The equation cannot be solved because it is undefined.

$Undefined$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(\frac{π}{2} + π n, 0)$ , for any integer $n$

y-intercept(s): $None$

Step 4