Trigonometry Examples

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

Step 1

Multiply the numerator and denominator of $\frac{- 3 + 4 i}{4 - 3 i}$ by the conjugate of $4 - 3 i$ to make the denominator real.

$\frac{- 3 + 4 i}{4 - 3 i} \cdot \frac{4 + 3 i}{4 + 3 i}$

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(- 3 + 4 i) (4 + 3 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $- 3$ by $4$ .

$\frac{- 12 - 3 (3 i) + 4 i \cdot 4 + 4 i (3 i)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.1.2

Multiply $3$ by $- 3$ .

$\frac{- 12 - 9 i + 4 i \cdot 4 + 4 i (3 i)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.1.3

Multiply $4$ by $4$ .

$\frac{- 12 - 9 i + 16 i + 4 i (3 i)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.1.4

Multiply $4 i (3 i)$ .

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Step 2.2.2.1.4.1

Multiply $3$ by $4$ .

$\frac{- 12 - 9 i + 16 i + 12 i i}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.1.4.2

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

$\frac{- 12 - 9 i + 16 i + 12 (i^{1} i)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.1.4.3

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

$\frac{- 12 - 9 i + 16 i + 12 (i^{1} i^{1})}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.1.4.4

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

$\frac{- 12 - 9 i + 16 i + 12 i^{1 + 1}}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.1.4.5

Add $1$ and $1$ .

$\frac{- 12 - 9 i + 16 i + 12 i^{2}}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.1.5

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

$\frac{- 12 - 9 i + 16 i + 12 \cdot - 1}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.1.6

Multiply $12$ by $- 1$ .

$\frac{- 12 - 9 i + 16 i - 12}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.2

Subtract $12$ from $- 12$ .

$\frac{- 24 - 9 i + 16 i}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.3

Add $- 9 i$ and $16 i$ .

$\frac{- 24 + 7 i}{(4 - 3 i) (4 + 3 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(4 - 3 i) (4 + 3 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 24 + 7 i}{4 (4 + 3 i) - 3 i (4 + 3 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 24 + 7 i}{4 \cdot 4 + 4 (3 i) - 3 i (4 + 3 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 24 + 7 i}{4 \cdot 4 + 4 (3 i) - 3 i \cdot 4 - 3 i (3 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $4$ by $4$ .

$\frac{- 24 + 7 i}{16 + 4 (3 i) - 3 i \cdot 4 - 3 i (3 i)}$

Step 2.3.2.2

Multiply $3$ by $4$ .

$\frac{- 24 + 7 i}{16 + 12 i - 3 i \cdot 4 - 3 i (3 i)}$

Step 2.3.2.3

Multiply $4$ by $- 3$ .

$\frac{- 24 + 7 i}{16 + 12 i - 12 i - 3 i (3 i)}$

Step 2.3.2.4

Multiply $3$ by $- 3$ .

$\frac{- 24 + 7 i}{16 + 12 i - 12 i - 9 i i}$

Step 2.3.2.5

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

$\frac{- 24 + 7 i}{16 + 12 i - 12 i - 9 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 24 + 7 i}{16 + 12 i - 12 i - 9 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 24 + 7 i}{16 + 12 i - 12 i - 9 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 24 + 7 i}{16 + 12 i - 12 i - 9 i^{2}}$

Step 2.3.2.9

Subtract $12 i$ from $12 i$ .

$\frac{- 24 + 7 i}{16 + 0 - 9 i^{2}}$

Step 2.3.2.10

Add $16$ and $0$ .

$\frac{- 24 + 7 i}{16 - 9 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 24 + 7 i}{16 - 9 \cdot - 1}$

Step 2.3.3.2

Multiply $- 9$ by $- 1$ .

$\frac{- 24 + 7 i}{16 + 9}$

Step 2.3.4

Add $16$ and $9$ .

$\frac{- 24 + 7 i}{25}$

Step 3

Split the fraction $\frac{- 24 + 7 i}{25}$ into two fractions.

$\frac{- 24}{25} + \frac{7 i}{25}$

Step 4

Move the negative in front of the fraction.

$- \frac{24}{25} + \frac{7 i}{25}$