Trigonometry Examples

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

Step 1

Multiply the numerator and denominator of $\frac{- 6 i}{3 + 2 i}$ by the conjugate of $3 + 2 i$ to make the denominator real.

$\frac{- 6 i}{3 + 2 i} \cdot \frac{3 - 2 i}{3 - 2 i}$

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $3$ by $- 6$ .

$\frac{- 18 i - 6 i (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.3

Multiply $- 6 i (- 2 i)$ .

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Step 2.2.3.1

Multiply $- 2$ by $- 6$ .

$\frac{- 18 i + 12 i i}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.3.2

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

$\frac{- 18 i + 12 (i^{1} i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.3.3

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

$\frac{- 18 i + 12 (i^{1} i^{1})}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.3.4

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

$\frac{- 18 i + 12 i^{1 + 1}}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.3.5

Add $1$ and $1$ .

$\frac{- 18 i + 12 i^{2}}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

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

$\frac{- 18 i + 12 \cdot - 1}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.4.2

Multiply $12$ by $- 1$ .

$\frac{- 18 i - 12}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.5

Reorder $- 18 i$ and $- 12$ .

$\frac{- 12 - 18 i}{(3 + 2 i) (3 - 2 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(3 + 2 i) (3 - 2 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 12 - 18 i}{3 (3 - 2 i) + 2 i (3 - 2 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 12 - 18 i}{3 \cdot 3 + 3 (- 2 i) + 2 i (3 - 2 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 12 - 18 i}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $3$ by $3$ .

$\frac{- 12 - 18 i}{9 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.3.2.2

Multiply $- 2$ by $3$ .

$\frac{- 12 - 18 i}{9 - 6 i + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.3.2.3

Multiply $3$ by $2$ .

$\frac{- 12 - 18 i}{9 - 6 i + 6 i + 2 i (- 2 i)}$

Step 2.3.2.4

Multiply $- 2$ by $2$ .

$\frac{- 12 - 18 i}{9 - 6 i + 6 i - 4 i i}$

Step 2.3.2.5

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

$\frac{- 12 - 18 i}{9 - 6 i + 6 i - 4 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 12 - 18 i}{9 - 6 i + 6 i - 4 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 12 - 18 i}{9 - 6 i + 6 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 12 - 18 i}{9 - 6 i + 6 i - 4 i^{2}}$

Step 2.3.2.9

Add $- 6 i$ and $6 i$ .

$\frac{- 12 - 18 i}{9 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add $9$ and $0$ .

$\frac{- 12 - 18 i}{9 - 4 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 12 - 18 i}{9 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{- 12 - 18 i}{9 + 4}$

Step 2.3.4

Add $9$ and $4$ .

$\frac{- 12 - 18 i}{13}$

Step 3

Split the fraction $\frac{- 12 - 18 i}{13}$ into two fractions.

$\frac{- 12}{13} + \frac{- 18 i}{13}$

Step 4

Simplify each term.

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Step 4.1

Move the negative in front of the fraction.

$- \frac{12}{13} + \frac{- 18 i}{13}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{12}{13} - \frac{18 i}{13}$