Trigonometry Examples

$\csc^{2} (x) + 3 \cot (x) = - 2$

Step 1

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

$(\cot^{2} (x) + 1) + 3 \cot (x) = - 2$

Step 2

Reorder the polynomial.

$\cot^{2} (x) + 3 \cot (x) + 1 = - 2$

Step 3

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

${(u)}^{2} + 3 (u) + 1 = - 2$

Step 4

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

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Step 4.1

Add $2$ to both sides of the equation.

$u^{2} + 3 u + 1 + 2 = 0$

Step 4.2

Add $1$ and $2$ .

$u^{2} + 3 u + 3 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = 3$ , and $c = 3$ into the quadratic formula and solve for $u$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $3$ to the power of $2$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 3}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $3$ .

$u = \frac{- 3 \pm \sqrt{9 - 12}}{2 \cdot 1}$

Step 7.1.3

Subtract $12$ from $9$ .

$u = \frac{- 3 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 7.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$u = \frac{- 3 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$u = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{- 3 \pm i \sqrt{3}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$u = \frac{- 3 \pm i \sqrt{3}}{2}$

Step 8

The final answer is the combination of both solutions.

$u = - \frac{3 - i \sqrt{3}}{2}, - \frac{3 + i \sqrt{3}}{2}$

Step 9

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

$\cot (x) = - \frac{3 - i \sqrt{3}}{2}, - \frac{3 + i \sqrt{3}}{2}$

Step 10

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

$\cot (x) = - \frac{3 - i \sqrt{3}}{2}$

$\cot (x) = - \frac{3 + i \sqrt{3}}{2}$

Step 11

Solve for $x$ in $\cot (x) = - \frac{3 - i \sqrt{3}}{2}$ .

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Step 11.1

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

$x = arccot (- \frac{3 - i \sqrt{3}}{2})$

Step 11.2

The inverse cotangent of $arccot (- \frac{3 - i \sqrt{3}}{2})$ is undefined.

Undefined

Step 12

Solve for $x$ in $\cot (x) = - \frac{3 + i \sqrt{3}}{2}$ .

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Step 12.1

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

$x = arccot (- \frac{3 + i \sqrt{3}}{2})$

Step 12.2

The inverse cotangent of $arccot (- \frac{3 + i \sqrt{3}}{2})$ is undefined.

Undefined

Step 13

List all of the solutions.

No solution