Trigonometry Examples

$\cot^{2} (x) = - \frac{5}{2} \cdot \csc (x) - 2$

Step 1

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

$\csc^{2} (x) - 1 = - \frac{5}{2} \cdot \csc (x) - 2$

Step 2

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

${(u)}^{2} - 1 = - \frac{5}{2} u - 2$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Combine $u$ and $\frac{5}{2}$ .

$u^{2} - 1 = - \frac{u \cdot 5}{2} - 2$

Step 3.2

Move $5$ to the left of $u$ .

$u^{2} - 1 = - \frac{5 u}{2} - 2$

Step 4

Add $\frac{5 u}{2}$ to both sides of the equation.

$u^{2} - 1 + \frac{5 u}{2} = - 2$

Step 5

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 5.1

Add $2$ to both sides of the equation.

$u^{2} - 1 + \frac{5 u}{2} + 2 = 0$

Step 5.2

Add $- 1$ and $2$ .

$u^{2} + \frac{5 u}{2} + 1 = 0$

Step 6

Multiply through by the least common denominator $2$ , then simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

$2 u^{2} + 2 \frac{5 u}{2} + 2 \cdot 1 = 0$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$2 u^{2} + 2 \frac{5 u}{2} + 2 \cdot 1 = 0$

Step 6.2.1.2

Rewrite the expression.

$2 u^{2} + 5 u + 2 \cdot 1 = 0$

Step 6.2.2

Multiply $2$ by $1$ .

$2 u^{2} + 5 u + 2 = 0$

Step 7

Use the quadratic formula to find the solutions.

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

Step 8

Substitute the values $a = 2$ , $b = 5$ , and $c = 2$ into the quadratic formula and solve for $u$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (2 \cdot 2)}}{2 \cdot 2}$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Simplify the numerator.

Tap for more steps...

Step 9.1.1

Raise $5$ to the power of $2$ .

$u = \frac{- 5 \pm \sqrt{25 - 4 \cdot 2 \cdot 2}}{2 \cdot 2}$

Step 9.1.2

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

Tap for more steps...

Step 9.1.2.1

Multiply $- 4$ by $2$ .

$u = \frac{- 5 \pm \sqrt{25 - 8 \cdot 2}}{2 \cdot 2}$

Step 9.1.2.2

Multiply $- 8$ by $2$ .

$u = \frac{- 5 \pm \sqrt{25 - 16}}{2 \cdot 2}$

Step 9.1.3

Subtract $16$ from $25$ .

$u = \frac{- 5 \pm \sqrt{9}}{2 \cdot 2}$

Step 9.1.4

Rewrite $9$ as $3^{2}$ .

$u = \frac{- 5 \pm \sqrt{3^{2}}}{2 \cdot 2}$

Step 9.1.5

Pull terms out from under the radical, assuming positive real numbers.

$u = \frac{- 5 \pm 3}{2 \cdot 2}$

Step 9.2

Multiply $2$ by $2$ .

$u = \frac{- 5 \pm 3}{4}$

Step 10

The final answer is the combination of both solutions.

$u = - \frac{1}{2}, - 2$

Step 11

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

$\csc (x) = - \frac{1}{2}, - 2$

Step 12

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

$\csc (x) = - \frac{1}{2}$

$\csc (x) = - 2$

Step 13

Solve for $x$ in $\csc (x) = - \frac{1}{2}$ .

Tap for more steps...

Step 13.1

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $- \frac{1}{2}$ does not fall in this range, there is no solution.

No solution

Step 14

Solve for $x$ in $\csc (x) = - 2$ .

Tap for more steps...

Step 14.1

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

$x = arccsc (- 2)$

Step 14.2

Simplify the right side.

Tap for more steps...

Step 14.2.1

The exact value of $arccsc (- 2)$ is $- \frac{π}{6}$ .

$x = - \frac{π}{6}$

Step 14.3

The cosecant function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{6} + π$

Step 14.4

Simplify the expression to find the second solution.

Tap for more steps...

Step 14.4.1

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$x = 2 π + \frac{π}{6} + π - 2 π$

Step 14.4.2

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$x = \frac{7 π}{6}$

Step 14.5

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

Tap for more steps...

Step 14.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 14.5.2

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

$\frac{2 π}{| 1 |}$

Step 14.5.3

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

$\frac{2 π}{1}$

Step 14.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 14.6

Add $2 π$ to every negative angle to get positive angles.

Tap for more steps...

Step 14.6.1

Add $2 π$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

Step 14.6.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 14.6.3

Combine fractions.

Tap for more steps...

Step 14.6.3.1

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 14.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Step 14.6.4

Simplify the numerator.

Tap for more steps...

Step 14.6.4.1

Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Step 14.6.4.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

Step 14.6.5

List the new angles.

$x = \frac{11 π}{6}$

Step 14.7

The period of the $\csc (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Step 15

List all of the solutions.

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$