Trigonometry Examples

$6 \cot (x + 5) = - 1$

Step 1

Divide each term in $6 \cot (x + 5) = - 1$ by $6$ and simplify.

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Step 1.1

Divide each term in $6 \cot (x + 5) = - 1$ by $6$ .

$\frac{6 \cot (x + 5)}{6} = \frac{- 1}{6}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $6$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{6 \cot (x + 5)}{6} = \frac{- 1}{6}$

Step 1.2.1.2

Divide $\cot (x + 5)$ by $1$ .

$\cot (x + 5) = \frac{- 1}{6}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$\cot (x + 5) = - \frac{1}{6}$

Step 2

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

$x + 5 = arccot (- \frac{1}{6})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $arccot (- \frac{1}{6})$ .

$x + 5 = 1.735945$

Step 4

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

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Step 4.1

Subtract $5$ from both sides of the equation.

$x = 1.735945 - 5$

Step 4.2

Subtract $5$ from $1.735945$ .

$x = - 3.26405499$

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.

$x + 5 = 1.735945 - (3.14159265)$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Add $2 π$ to $1.735945 - (3.14159265)$ .

$x + 5 = 1.735945 - (3.14159265) + 2 π$

Step 6.2

The resulting angle of $4.87753765$ is positive and coterminal with $1.735945 - (3.14159265)$ .

$x + 5 = 4.87753765$

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 $5$ from both sides of the equation.

$x = 4.87753765 - 5$

Step 6.3.2

Subtract $5$ from $4.87753765$ .

$x = - 0.12246234$

Step 7

Find the period of $\cot (x + 5)$ .

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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

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

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Step 8.1

Add $π$ to $- 3.26405499$ to find the positive angle.

$- 3.26405499 + π$

Step 8.2

Replace with decimal approximation.

$3.14159265 - 3.26405499$

Step 8.3

Subtract $3.26405499$ from $3.14159265$ .

$- 0.12246234$

Step 8.4

Add $π$ to $- 0.12246234$ to find the positive angle.

$- 0.12246234 + π$

Step 8.5

Replace with decimal approximation.

$3.14159265 - 0.12246234$

Step 8.6

Subtract $0.12246234$ from $3.14159265$ .

$3.01913031$

Step 8.7

List the new angles.

$x = 3.01913031$

Step 9

The period of the $\cot (x + 5)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = - 0.12246234 + π n, 3.01913031 + π n$ , for any integer $n$