Trigonometry Examples

$\cot^{2} (x) - 8 \cot (x) + 12 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = \cot (x)$ . Substitute $u$ for all occurrences of $\cot (x)$ .

$u^{2} - 8 u + 12 = 0$

Step 1.2

Factor $u^{2} - 8 u + 12$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $- 8$ .

$- 6, - 2$

Step 1.2.2

Write the factored form using these integers.

$(u - 6) (u - 2) = 0$

Step 1.3

Replace all occurrences of $u$ with $\cot (x)$ .

$(\cot (x) - 6) (\cot (x) - 2) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cot (x) - 6 = 0$

$\cot (x) - 2 = 0$

Step 3

Set $\cot (x) - 6$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $\cot (x) - 6$ equal to $0$ .

$\cot (x) - 6 = 0$

Step 3.2

Solve $\cot (x) - 6 = 0$ for $x$ .

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Step 3.2.1

Add $6$ to both sides of the equation.

$\cot (x) = 6$

Step 3.2.2

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

$x = arccot (6)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Evaluate $arccot (6)$ .

$x = 0.16514867$

Step 3.2.4

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.

$x = (3.14159265) + 0.16514867$

Step 3.2.5

Solve for $x$ .

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Step 3.2.5.1

Remove parentheses.

$x = 3.14159265 + 0.16514867$

Step 3.2.5.2

Remove parentheses.

$x = (3.14159265) + 0.16514867$

Step 3.2.5.3

Add $3.14159265$ and $0.16514867$ .

$x = 3.30674133$

Step 3.2.6

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

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Step 3.2.6.1

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

$\frac{π}{| b |}$

Step 3.2.6.2

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

$\frac{π}{| 1 |}$

Step 3.2.6.3

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

$\frac{π}{1}$

Step 3.2.6.4

Divide $π$ by $1$ .

$π$

Step 3.2.7

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

$x = 0.16514867 + π n, 3.30674133 + π n$ , for any integer $n$

Step 4

Set $\cot (x) - 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\cot (x) - 2$ equal to $0$ .

$\cot (x) - 2 = 0$

Step 4.2

Solve $\cot (x) - 2 = 0$ for $x$ .

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Step 4.2.1

Add $2$ to both sides of the equation.

$\cot (x) = 2$

Step 4.2.2

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

$x = arccot (2)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Evaluate $arccot (2)$ .

$x = 0.4636476$

Step 4.2.4

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.

$x = (3.14159265) + 0.4636476$

Step 4.2.5

Solve for $x$ .

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Step 4.2.5.1

Remove parentheses.

$x = 3.14159265 + 0.4636476$

Step 4.2.5.2

Remove parentheses.

$x = (3.14159265) + 0.4636476$

Step 4.2.5.3

Add $3.14159265$ and $0.4636476$ .

$x = 3.60524026$

Step 4.2.6

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

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Step 4.2.6.1

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

$\frac{π}{| b |}$

Step 4.2.6.2

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

$\frac{π}{| 1 |}$

Step 4.2.6.3

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

$\frac{π}{1}$

Step 4.2.6.4

Divide $π$ by $1$ .

$π$

Step 4.2.7

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

$x = 0.4636476 + π n, 3.60524026 + π n$ , for any integer $n$

Step 5

The final solution is all the values that make $(\cot (x) - 6) (\cot (x) - 2) = 0$ true.

$x = 0.16514867 + π n, 3.30674133 + π n, 0.4636476 + π n, 3.60524026 + π n$ , for any integer $n$

Step 6

Consolidate the answers.

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Step 6.1

Consolidate $0.16514867 + π n$ and $3.30674133 + π n$ to $0.16514867 + π n$ .

$x = 0.16514867 + π n, 0.4636476 + π n, 3.60524026 + π n$ , for any integer $n$

Step 6.2

Consolidate $0.4636476 + π n$ and $3.60524026 + π n$ to $0.4636476 + π n$ .

$x = 0.16514867 + π n, 0.4636476 + π n$ , for any integer $n$