Precalculus Examples

$4 \cot (x + 2) = - 2$

Step 1

Divide each term in $4 \cot (x + 2) = - 2$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 \cot (x + 2) = - 2$ by $4$ .

$\frac{4 \cot (x + 2)}{4} = \frac{- 2}{4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{4 \cot (x + 2)}{4} = \frac{- 2}{4}$

Step 1.2.1.2

Divide $\cot (x + 2)$ by $1$ .

$\cot (x + 2) = \frac{- 2}{4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $- 2$ and $4$ .

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Step 1.3.1.1

Factor $2$ out of $- 2$ .

$\cot (x + 2) = \frac{2 (- 1)}{4}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2$ out of $4$ .

$\cot (x + 2) = \frac{2 \cdot - 1}{2 \cdot 2}$

Step 1.3.1.2.2

Cancel the common factor.

$\cot (x + 2) = \frac{2 \cdot - 1}{2 \cdot 2}$

Step 1.3.1.2.3

Rewrite the expression.

$\cot (x + 2) = \frac{- 1}{2}$

Step 1.3.2

Move the negative in front of the fraction.

$\cot (x + 2) = - \frac{1}{2}$

Step 2

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

$x + 2 = arccot (- \frac{1}{2})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $arccot (- \frac{1}{2})$ .

$x + 2 = 2.03444393$

Step 4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1

Subtract $2$ from both sides of the equation.

$x = 2.03444393 - 2$

Step 4.2

Subtract $2$ from $2.03444393$ .

$x = 0.03444393$

Step 5

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.

$x + 2 = 2.03444393 - (3.14159265)$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Add $2 π$ to $2.03444393 - (3.14159265)$ .

$x + 2 = 2.03444393 - (3.14159265) + 2 π$

Step 6.2

The resulting angle of $5.17603658$ is positive and coterminal with $2.03444393 - (3.14159265)$ .

$x + 2 = 5.17603658$

Step 6.3

Move all terms not containing $x$ to the right side of the equation.

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Step 6.3.1

Subtract $2$ from both sides of the equation.

$x = 5.17603658 - 2$

Step 6.3.2

Subtract $2$ from $5.17603658$ .

$x = 3.17603658$

Step 7

Find the period of $\cot (x + 2)$ .

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Step 7.1

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

$\frac{π}{| b |}$

Step 7.2

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

$\frac{π}{| 1 |}$

Step 7.3

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

$\frac{π}{1}$

Step 7.4

Divide $π$ by $1$ .

$π$

Step 8

The period of the $\cot (x + 2)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = 0.03444393 + π n, 3.17603658 + π n$ , for any integer $n$

Step 9

Consolidate $0.03444393 + π n$ and $3.17603658 + π n$ to $0.03444393 + π n$ .

$x = 0.03444393 + π n$ , for any integer $n$