Trigonometry Examples

$\cot^{2} (x) - 15 \cot (x) = 16$

Step 1

Subtract $16$ from both sides of the equation.

$\cot^{2} (x) - 15 \cot (x) - 16 = 0$

Step 2

Factor $\cot^{2} (x) - 15 \cot (x) - 16$ using the AC method.

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Step 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 $- 16$ and whose sum is $- 15$ .

$- 16, 1$

Step 2.2

Write the factored form using these integers.

$(\cot (x) - 16) (\cot (x) + 1) = 0$

Step 3

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

$\cot (x) - 16 = 0$

$\cot (x) + 1 = 0$

Step 4

Set $\cot (x) - 16$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\cot (x) - 16$ equal to $0$ .

$\cot (x) - 16 = 0$

Step 4.2

Solve $\cot (x) - 16 = 0$ for $x$ .

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Step 4.2.1

Add $16$ to both sides of the equation.

$\cot (x) = 16$

Step 4.2.2

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

$x = arccot (16)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Evaluate $arccot (16)$ .

$x = 0.0624188$

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.

$x = (3.14159265) + 0.0624188$

Step 4.2.5

Solve for $x$ .

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Step 4.2.5.1

Remove parentheses.

$x = 3.14159265 + 0.0624188$

Step 4.2.5.2

Remove parentheses.

$x = (3.14159265) + 0.0624188$

Step 4.2.5.3

Add $3.14159265$ and $0.0624188$ .

$x = 3.20401146$

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.

$x = 0.0624188 + π n, 3.20401146 + π n$ , for any integer $n$

Step 5

Set $\cot (x) + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $\cot (x) + 1$ equal to $0$ .

$\cot (x) + 1 = 0$

Step 5.2

Solve $\cot (x) + 1 = 0$ for $x$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$\cot (x) = - 1$

Step 5.2.2

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

$x = arccot (- 1)$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

The exact value of $arccot (- 1)$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 5.2.4

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.

$x = \frac{3 π}{4} - π$

Step 5.2.5

Simplify the expression to find the second solution.

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Step 5.2.5.1

Add $2 π$ to $\frac{3 π}{4} - π$ .

$x = \frac{3 π}{4} - π + 2 π$

Step 5.2.5.2

The resulting angle of $\frac{7 π}{4}$ is positive and coterminal with $\frac{3 π}{4} - π$ .

$x = \frac{7 π}{4}$

Step 5.2.6

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

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Step 5.2.6.1

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

$\frac{π}{| b |}$

Step 5.2.6.2

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

$\frac{π}{| 1 |}$

Step 5.2.6.3

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

$\frac{π}{1}$

Step 5.2.6.4

Divide $π$ by $1$ .

$π$

Step 5.2.7

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

$x = \frac{3 π}{4} + π n, \frac{7 π}{4} + π n$ , for any integer $n$

Step 6

The final solution is all the values that make $(\cot (x) - 16) (\cot (x) + 1) = 0$ true.

$x = 0.0624188 + π n, 3.20401146 + π n, \frac{3 π}{4} + π n, \frac{7 π}{4} + π n$ , for any integer $n$

Step 7

Consolidate the answers.

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Step 7.1

Consolidate $0.0624188 + π n$ and $3.20401146 + π n$ to $0.0624188 + π n$ .

$x = 0.0624188 + π n, \frac{3 π}{4} + π n, \frac{7 π}{4} + π n$ , for any integer $n$

Step 7.2

Consolidate $\frac{3 π}{4} + π n$ and $\frac{7 π}{4} + π n$ to $\frac{3 π}{4} + π n$ .

$x = 0.0624188 + π n, \frac{3 π}{4} + π n$ , for any integer $n$