Trigonometry Examples

$\cot^{2} (x) + 6 \cot (x) - 2 = 0$

Step 1

Substitute $u$ for $\cot (x)$ .

${(u)}^{2} + 6 (u) - 2 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = 6$ , and $c = - 2$ into the quadratic formula and solve for $u$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot - 2)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $6$ to the power of $2$ .

$u = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 2}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 2}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 2$ .

$u = \frac{- 6 \pm \sqrt{36 + 8}}{2 \cdot 1}$

Step 4.1.3

Add $36$ and $8$ .

$u = \frac{- 6 \pm \sqrt{44}}{2 \cdot 1}$

Step 4.1.4

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 4.1.4.1

Factor $4$ out of $44$ .

$u = \frac{- 6 \pm \sqrt{4 (11)}}{2 \cdot 1}$

Step 4.1.4.2

Rewrite $4$ as $2^{2}$ .

$u = \frac{- 6 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

$u = \frac{- 6 \pm 2 \sqrt{11}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$u = \frac{- 6 \pm 2 \sqrt{11}}{2}$

Step 4.3

Simplify $\frac{- 6 \pm 2 \sqrt{11}}{2}$ .

$u = - 3 \pm \sqrt{11}$

Step 5

The final answer is the combination of both solutions.

$u = - 3 + \sqrt{11}, - 3 - \sqrt{11}$

Step 6

Substitute $\cot (x)$ for $u$ .

$\cot (x) = - 3 + \sqrt{11}, - 3 - \sqrt{11}$

Step 7

Set up each of the solutions to solve for $x$ .

$\cot (x) = - 3 + \sqrt{11}$

$\cot (x) = - 3 - \sqrt{11}$

Step 8

Solve for $x$ in $\cot (x) = - 3 + \sqrt{11}$ .

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Step 8.1

Convert the right side of the equation to its decimal equivalent.

$\cot (x) = 0.31662479$

Step 8.2

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

$x = arccot (0.31662479)$

Step 8.3

Simplify the right side.

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Step 8.3.1

Evaluate $arccot (0.31662479)$ .

$x = 1.26415806$

Step 8.4

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.

$x = (3.14159265) + 1.26415806$

Step 8.5

Solve for $x$ .

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Step 8.5.1

Remove parentheses.

$x = 3.14159265 + 1.26415806$

Step 8.5.2

Remove parentheses.

$x = (3.14159265) + 1.26415806$

Step 8.5.3

Add $3.14159265$ and $1.26415806$ .

$x = 4.40575072$

Step 8.6

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

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Step 8.6.1

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

$\frac{π}{| b |}$

Step 8.6.2

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

$\frac{π}{| 1 |}$

Step 8.6.3

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

$\frac{π}{1}$

Step 8.6.4

Divide $π$ by $1$ .

$π$

Step 8.7

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

$x = 1.26415806 + π n, 4.40575072 + π n$ , for any integer $n$

Step 9

Solve for $x$ in $\cot (x) = - 3 - \sqrt{11}$ .

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Step 9.1

Convert the right side of the equation to its decimal equivalent.

$\cot (x) = - 6.31662479$

Step 9.2

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

$x = arccot (- 6.31662479)$

Step 9.3

Simplify the right side.

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Step 9.3.1

Evaluate $arccot (- 6.31662479)$ .

$x = 2.9845833$

Step 9.4

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.

$x = 2.9845833 - (3.14159265)$

Step 9.5

Simplify the expression to find the second solution.

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Step 9.5.1

Add $2 π$ to $2.9845833 - (3.14159265)$ .

$x = 2.9845833 - (3.14159265) + 2 π$

Step 9.5.2

The resulting angle of $6.12617595$ is positive and coterminal with $2.9845833 - (3.14159265)$ .

$x = 6.12617595$

Step 9.6

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

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Step 9.6.1

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

$\frac{π}{| b |}$

Step 9.6.2

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

$\frac{π}{| 1 |}$

Step 9.6.3

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

$\frac{π}{1}$

Step 9.6.4

Divide $π$ by $1$ .

$π$

Step 9.7

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

$x = 2.9845833 + π n, 6.12617595 + π n$ , for any integer $n$

Step 10

List all of the solutions.

$x = 1.26415806 + π n, 4.40575072 + π n, 2.9845833 + π n, 6.12617595 + π n$ , for any integer $n$

Step 11

Consolidate the solutions.

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Step 11.1

Consolidate $1.26415806 + π n$ and $4.40575072 + π n$ to $1.26415806 + π n$ .

$x = 1.26415806 + π n, 2.9845833 + π n, 6.12617595 + π n$ , for any integer $n$

Step 11.2

Consolidate $2.9845833 + π n$ and $6.12617595 + π n$ to $2.9845833 + π n$ .

$x = 1.26415806 + π n, 2.9845833 + π n$ , for any integer $n$