Trigonometry Examples

$\frac{- 6 - \sqrt{- 18}}{6}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 18$ as $- 1 (18)$ .

$\frac{- 6 - \sqrt{- 1 (18)}}{6}$

Step 1.2

Rewrite $\sqrt{- 1 (18)}$ as $\sqrt{- 1} \cdot \sqrt{18}$ .

$\frac{- 6 - (\sqrt{- 1} \cdot \sqrt{18})}{6}$

Step 1.3

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

$\frac{- 6 - (i \cdot \sqrt{18})}{6}$

$\frac{- 6 - (i \cdot \sqrt{18})}{6}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.1.1

Factor $9$ out of $18$ .

$\frac{- 6 - i \cdot \sqrt{9 (2)}}{6}$

Step 2.1.2

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

$\frac{- 6 - i \cdot \sqrt{3^{2} \cdot 2}}{6}$

$\frac{- 6 - i \cdot \sqrt{3^{2} \cdot 2}}{6}$

Step 2.2

Pull terms out from under the radical.

$\frac{- 6 - i \cdot (3 \sqrt{2})}{6}$

Step 2.3

Multiply $3$ by $- 1$ .

$\frac{- 6 - 3 i \sqrt{2}}{6}$

$\frac{- 6 - 3 i \sqrt{2}}{6}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $- 6 - 3 i \sqrt{2}$ and $6$ .

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Step 3.1.1

Factor $3$ out of $- 6$ .

$\frac{3 (- 2) - 3 i \sqrt{2}}{6}$

Step 3.1.2

Factor $3$ out of $- 3 i \sqrt{2}$ .

$\frac{3 (- 2) + 3 (- i \sqrt{2})}{6}$

Step 3.1.3

Factor $3$ out of $3 (- 2) + 3 (- i \sqrt{2})$ .

$\frac{3 (- 2 - i \sqrt{2})}{6}$

Step 3.1.4

Cancel the common factors.

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Step 3.1.4.1

Factor $3$ out of $6$ .

$\frac{3 (- 2 - i \sqrt{2})}{3 \cdot 2}$

Step 3.1.4.2

Cancel the common factor.

$\frac{3 (- 2 - i \sqrt{2})}{3 \cdot 2}$

Step 3.1.4.3

Rewrite the expression.

$\frac{- 2 - i \sqrt{2}}{2}$

$\frac{- 2 - i \sqrt{2}}{2}$

$\frac{- 2 - i \sqrt{2}}{2}$

Step 3.2

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- 1 (2) - i \sqrt{2}}{2}$

Step 3.3

Factor $- 1$ out of $- i \sqrt{2}$ .

$\frac{- 1 (2) - (i \sqrt{2})}{2}$

Step 3.4

Factor $- 1$ out of $- 1 (2) - (i \sqrt{2})$ .

$\frac{- 1 (2 + i \sqrt{2})}{2}$

Step 3.5

Move the negative in front of the fraction.

$- \frac{2 + i \sqrt{2}}{2}$

$- \frac{2 + i \sqrt{2}}{2}$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$