Trigonometry Examples

$\frac{9 - 9 i}{\sqrt{3} + i}$

Step 1

Multiply $\frac{9 - 9 i}{\sqrt{3} + i}$ by $\frac{\sqrt{3} - i}{\sqrt{3} - i}$ .

$\frac{9 - 9 i}{\sqrt{3} + i} \cdot \frac{\sqrt{3} - i}{\sqrt{3} - i}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{9 - 9 i}{\sqrt{3} + i}$ by $\frac{\sqrt{3} - i}{\sqrt{3} - i}$ .

$\frac{(9 - 9 i) (\sqrt{3} - i)}{(\sqrt{3} + i) (\sqrt{3} - i)}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(9 - 9 i) (\sqrt{3} - i)}{{\sqrt{3}}^{2} + \sqrt{3} (- i) + i \sqrt{3} - i^{2}}$

Step 2.3

Simplify.

$\frac{(9 - 9 i) (\sqrt{3} - i)}{4}$

Step 3

Expand $(9 - 9 i) (\sqrt{3} - i)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{9 (\sqrt{3} - i) - 9 i (\sqrt{3} - i)}{4}$

Step 3.2

Apply the distributive property.

$\frac{9 \sqrt{3} + 9 (- i) - 9 i (\sqrt{3} - i)}{4}$

Step 3.3

Apply the distributive property.

$\frac{9 \sqrt{3} + 9 (- i) - 9 i \sqrt{3} - 9 i (- i)}{4}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $- 1$ by $9$ .

$\frac{9 \sqrt{3} - 9 i - 9 i \sqrt{3} - 9 i (- i)}{4}$

Step 4.1.2

Multiply $- 9 i (- i)$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 9$ .

$\frac{9 \sqrt{3} - 9 i - 9 i \sqrt{3} + 9 i i}{4}$

Step 4.1.2.2

Raise $i$ to the power of $1$ .

$\frac{9 \sqrt{3} - 9 i - 9 i \sqrt{3} + 9 (i^{1} i)}{4}$

Step 4.1.2.3

Raise $i$ to the power of $1$ .

$\frac{9 \sqrt{3} - 9 i - 9 i \sqrt{3} + 9 (i^{1} i^{1})}{4}$

Step 4.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 \sqrt{3} - 9 i - 9 i \sqrt{3} + 9 i^{1 + 1}}{4}$

Step 4.1.2.5

Add $1$ and $1$ .

$\frac{9 \sqrt{3} - 9 i - 9 i \sqrt{3} + 9 i^{2}}{4}$

Step 4.1.3

Rewrite $i^{2}$ as $- 1$ .

$\frac{9 \sqrt{3} - 9 i - 9 i \sqrt{3} + 9 \cdot - 1}{4}$

Step 4.1.4

Multiply $9$ by $- 1$ .

$\frac{9 \sqrt{3} - 9 i - 9 i \sqrt{3} - 9}{4}$

Step 4.2

Simplify with factoring out.

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Step 4.2.1

Reorder $9 \sqrt{3} - 9 i - 9 i \sqrt{3}$ and $- 9$ .

$\frac{- 9 + 9 \sqrt{3} - 9 i - 9 i \sqrt{3}}{4}$

Step 4.2.2

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{- 1 (9) + 9 \sqrt{3} - 9 i - 9 i \sqrt{3}}{4}$

Step 4.2.3

Factor $- 1$ out of $9 \sqrt{3}$ .

$\frac{- 1 (9) - (- 9 \sqrt{3}) - 9 i - 9 i \sqrt{3}}{4}$

Step 4.2.4

Factor $- 1$ out of $- 1 (9) - (- 9 \sqrt{3})$ .

$\frac{- 1 (9 - 9 \sqrt{3}) - 9 i - 9 i \sqrt{3}}{4}$

Step 4.2.5

Factor $- 1$ out of $- 9 i$ .

$\frac{- 1 (9 - 9 \sqrt{3}) - (9 i) - 9 i \sqrt{3}}{4}$

Step 4.2.6

Factor $- 1$ out of $- 1 (9 - 9 \sqrt{3}) - (9 i)$ .

$\frac{- 1 (9 - 9 \sqrt{3} + 9 i) - 9 i \sqrt{3}}{4}$

Step 4.2.7

Factor $- 1$ out of $- 9 i \sqrt{3}$ .

$\frac{- 1 (9 - 9 \sqrt{3} + 9 i) - (9 i \sqrt{3})}{4}$

Step 4.2.8

Factor $- 1$ out of $- 1 (9 - 9 \sqrt{3} + 9 i) - (9 i \sqrt{3})$ .

$\frac{- 1 (9 - 9 \sqrt{3} + 9 i + 9 i \sqrt{3})}{4}$

Step 4.2.9

Move the negative in front of the fraction.

$- \frac{9 - 9 \sqrt{3} + 9 i + 9 i \sqrt{3}}{4}$