Trigonometry Examples

$\cot^{2} (θ) - 6 \cot (θ) + 8 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

$u^{2} - 6 u + 8 = 0$

Step 1.2

Factor $u^{2} - 6 u + 8$ 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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 1.2.2

Write the factored form using these integers.

$(u - 4) (u - 2) = 0$

Step 1.3

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

$(\cot (θ) - 4) (\cot (θ) - 2) = 0$

Step 2

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

$\cot (θ) - 4 = 0$

$\cot (θ) - 2 = 0$

Step 3

Set $\cot (θ) - 4$ equal to $0$ and solve for $θ$ .

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Step 3.1

Set $\cot (θ) - 4$ equal to $0$ .

$\cot (θ) - 4 = 0$

Step 3.2

Solve $\cot (θ) - 4 = 0$ for $θ$ .

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Step 3.2.1

Add $4$ to both sides of the equation.

$\cot (θ) = 4$

Step 3.2.2

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

$θ = arccot (4)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Evaluate $arccot (4)$ .

$θ = 14.03624346$

Step 3.2.4

The cotangent function is positive in the first and third quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the fourth quadrant.

$θ = 180 + 14.03624346$

Step 3.2.5

Add $180$ and $14.03624346$ .

$θ = 194.03624346$

Step 3.2.6

Find the period of $\cot (θ)$ .

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Step 3.2.6.1

The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

Step 3.2.6.2

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

$\frac{180}{| 1 |}$

Step 3.2.6.3

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

$\frac{180}{1}$

Step 3.2.6.4

Divide $180$ by $1$ .

$180$

Step 3.2.7

The period of the $\cot (θ)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$θ = 14.03624346 + 180 n, 194.03624346 + 180 n$ , for any integer $n$

Step 4

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

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Step 4.1

Set $\cot (θ) - 2$ equal to $0$ .

$\cot (θ) - 2 = 0$

Step 4.2

Solve $\cot (θ) - 2 = 0$ for $θ$ .

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Step 4.2.1

Add $2$ to both sides of the equation.

$\cot (θ) = 2$

Step 4.2.2

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

$θ = arccot (2)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Evaluate $arccot (2)$ .

$θ = 26.56505117$

Step 4.2.4

The cotangent function is positive in the first and third quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the fourth quadrant.

$θ = 180 + 26.56505117$

Step 4.2.5

Add $180$ and $26.56505117$ .

$θ = 206.56505117$

Step 4.2.6

Find the period of $\cot (θ)$ .

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Step 4.2.6.1

The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

Step 4.2.6.2

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

$\frac{180}{| 1 |}$

Step 4.2.6.3

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

$\frac{180}{1}$

Step 4.2.6.4

Divide $180$ by $1$ .

$180$

Step 4.2.7

The period of the $\cot (θ)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$θ = 26.56505117 + 180 n, 206.56505117 + 180 n$ , for any integer $n$

Step 5

The final solution is all the values that make $(\cot (θ) - 4) (\cot (θ) - 2) = 0$ true.

$θ = 14.03624346 + 180 n, 194.03624346 + 180 n, 26.56505117 + 180 n, 206.56505117 + 180 n$ , for any integer $n$

Step 6

Consolidate the answers.

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Step 6.1

Consolidate $14.03624346 + 180 n$ and $194.03624346 + 180 n$ to $14.03624346 + 180 n$ .

$θ = 14.03624346 + 180 n, 26.56505117 + 180 n, 206.56505117 + 180 n$ , for any integer $n$

Step 6.2

Consolidate $26.56505117 + 180 n$ and $206.56505117 + 180 n$ to $26.56505117 + 180 n$ .

$θ = 14.03624346 + 180 n, 26.56505117 + 180 n$ , for any integer $n$