Precalculus Examples

$\csc^{2} (x) = 6 \cot (x) + 8$

Step 1

Replace the $\csc^{2} (x)$ with $\cot^{2} (x) + 1$ based on the $\cot^{2} (x) + 1 = \csc^{2} (x)$ identity.

$\cot^{2} (x) + 1 = 6 \cot (x) + 8$

Step 2

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

${(u)}^{2} + 1 = 6 (u) + 8$

Step 3

Subtract $6 u$ from both sides of the equation.

$u^{2} + 1 - 6 u = 8$

Step 4

Subtract $8$ from both sides of the equation.

$u^{2} + 1 - 6 u - 8 = 0$

Step 5

Subtract $8$ from $1$ .

$u^{2} - 6 u - 7 = 0$

Step 6

Factor $u^{2} - 6 u - 7$ using the AC method.

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Step 6.1

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 $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 6.2

Write the factored form using these integers.

$(u - 7) (u + 1) = 0$

Step 7

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

$u - 7 = 0$

$u + 1 = 0$

Step 8

Set $u - 7$ equal to $0$ and solve for $u$ .

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Step 8.1

Set $u - 7$ equal to $0$ .

$u - 7 = 0$

Step 8.2

Add $7$ to both sides of the equation.

$u = 7$

Step 9

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 9.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 9.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 10

The final solution is all the values that make $(u - 7) (u + 1) = 0$ true.

$u = 7, - 1$

Step 11

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

$\cot (x) = 7, - 1$

Step 12

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

$\cot (x) = 7$

$\cot (x) = - 1$

Step 13

Solve for $x$ in $\cot (x) = 7$ .

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Step 13.1

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

$x = arccot (7)$

Step 13.2

Simplify the right side.

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Step 13.2.1

Evaluate $arccot (7)$ .

$x = 0.14189705$

Step 13.3

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.

$x = (3.14159265) + 0.14189705$

Step 13.4

Solve for $x$ .

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Step 13.4.1

Remove parentheses.

$x = 3.14159265 + 0.14189705$

Step 13.4.2

Remove parentheses.

$x = (3.14159265) + 0.14189705$

Step 13.4.3

Add $3.14159265$ and $0.14189705$ .

$x = 3.2834897$

Step 13.5

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

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Step 13.5.1

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

$\frac{π}{| b |}$

Step 13.5.2

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

$\frac{π}{| 1 |}$

Step 13.5.3

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

$\frac{π}{1}$

Step 13.5.4

Divide $π$ by $1$ .

$π$

Step 13.6

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

$x = 0.14189705 + π n, 3.2834897 + π n$ , for any integer $n$

Step 14

Solve for $x$ in $\cot (x) = - 1$ .

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Step 14.1

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

$x = arccot (- 1)$

Step 14.2

Simplify the right side.

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Step 14.2.1

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

$x = \frac{3 π}{4}$

Step 14.3

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 14.4

Simplify the expression to find the second solution.

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Step 14.4.1

Add $2 π$ to $\frac{3 π}{4} - π$ .

$x = \frac{3 π}{4} - π + 2 π$

Step 14.4.2

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

$x = \frac{7 π}{4}$

Step 14.5

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

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Step 14.5.1

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

$\frac{π}{| b |}$

Step 14.5.2

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

$\frac{π}{| 1 |}$

Step 14.5.3

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

$\frac{π}{1}$

Step 14.5.4

Divide $π$ by $1$ .

$π$

Step 14.6

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 15

List all of the solutions.

$x = 0.14189705 + π n, 3.2834897 + π n, \frac{3 π}{4} + π n, \frac{7 π}{4} + π n$ , for any integer $n$

Step 16

Consolidate the solutions.

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Step 16.1

Consolidate $0.14189705 + π n$ and $3.2834897 + π n$ to $0.14189705 + π n$ .

$x = 0.14189705 + π n, \frac{3 π}{4} + π n, \frac{7 π}{4} + π n$ , for any integer $n$

Step 16.2

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

$x = 0.14189705 + π n, \frac{3 π}{4} + π n$ , for any integer $n$