Algebra Examples

$\cot (x) = - \frac{\sqrt{3}}{3}$

Step 1

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $3$ .

$\cot (x) \cdot \frac{3}{3} = - \frac{\sqrt{3}}{3}$

Step 2

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $3$ .

$\cot (x) \cdot 3$

Step 3

Move $3$ to the left of $\cot (x)$ .

$\frac{3 \cot (x)}{3} = - \frac{\sqrt{3}}{3}$

Step 4

Simplify $\cot (x) \cdot \frac{3}{3}$ .

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Step 4.1

Divide $3$ by $3$ .

$\cot (x) \cdot 1 = - \frac{\sqrt{3}}{3}$

Step 4.2

Multiply $\cot (x)$ by $1$ .

$\cot (x) = - \frac{\sqrt{3}}{3}$

Step 5

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

$x = arccot (- \frac{\sqrt{3}}{3})$

Step 6

Simplify the right side.

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Step 6.1

The exact value of $arccot (- \frac{\sqrt{3}}{3})$ is $\frac{2 π}{3}$ .

$x = \frac{2 π}{3}$

Step 7

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

$x = \frac{2 π}{3} - π$

Step 8

Simplify the expression to find the second solution.

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Step 8.1

Add $2 π$ to $\frac{2 π}{3} - π$ .

$x = \frac{2 π}{3} - π + 2 π$

Step 8.2

The resulting angle of $\frac{5 π}{3}$ is positive and coterminal with $\frac{2 π}{3} - π$ .

$x = \frac{5 π}{3}$

Step 9

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

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Step 9.1

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

$\frac{π}{| b |}$

Step 9.2

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

$\frac{π}{| 1 |}$

Step 9.3

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

$\frac{π}{1}$

Step 9.4

Divide $π$ by $1$ .

$π$

Step 10

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

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

Step 11

Consolidate the answers.

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