Trigonometry Examples

$\frac{- 1 - i}{- 1 + i}$

Step 1

Multiply the numerator and denominator of $\frac{- 1 - i}{- 1 + 1 i}$ by the conjugate of $- 1 + 1 i$ to make the denominator real.

$\frac{- 1 - i}{- 1 + 1 i} \cdot \frac{- 1 - i}{- 1 - i}$

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(- 1 - i) (- 1 - i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

$\frac{- 1 \cdot - 1 - 1 (- i) - i (- 1 - i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{- 1 \cdot - 1 - 1 (- i) - i \cdot - 1 - i (- i)}{(- 1 + 1 i) (- 1 - i)}$

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 $- 1$ by $- 1$ .

$\frac{1 - 1 (- i) - i \cdot - 1 - i (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.2

Multiply $- 1$ by $- 1$ .

$\frac{1 + 1 i - i \cdot - 1 - i (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.3

Multiply $i$ by $1$ .

$\frac{1 + i - i \cdot - 1 - i (- i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.4

Multiply $- 1$ by $- 1$ .

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

Step 2.2.2.1.5

Multiply $i$ by $1$ .

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

Step 2.2.2.1.6

Multiply $- i (- i)$ .

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Step 2.2.2.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.2.1.6.2

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

$\frac{1 + i + i + 1 (i^{1} i)}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.6.3

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

$\frac{1 + i + i + 1 (i^{1} i^{1})}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.6.4

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

$\frac{1 + i + i + 1 i^{1 + 1}}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.1.6.5

Add $1$ and $1$ .

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

Step 2.2.2.1.6.6

Multiply $i^{2}$ by $1$ .

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

Step 2.2.2.1.7

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

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

Step 2.2.2.2

Subtract $1$ from $1$ .

$\frac{0 + i + i}{(- 1 + 1 i) (- 1 - i)}$

Step 2.2.2.3

Add $0$ and $i$ .

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

Step 2.2.2.4

Add $i$ and $i$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(- 1 + 1 i) (- 1 - i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.2.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3.2.3

Multiply $- 1$ by $1$ .

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

Step 2.3.2.4

Multiply $- 1$ by $1$ .

$\frac{2 i}{1 + 1 i - i - i i}$

Step 2.3.2.5

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

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

Step 2.3.2.6

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

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

Step 2.3.2.7

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

$\frac{2 i}{1 + 1 i - i - i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{2 i}{1 + 1 i - i - i^{2}}$

Step 2.3.2.9

Subtract $i$ from $1 i$ .

$\frac{2 i}{1 + 0 - i^{2}}$

Step 2.3.2.10

Add $1$ and $0$ .

$\frac{2 i}{1 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{2 i}{1 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{2 i}{1 + 1}$

Step 2.3.4

Add $1$ and $1$ .

$\frac{2 i}{2}$

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Cancel the common factor.

$\frac{2 i}{2}$

Step 3.2

Divide $i$ by $1$ .

$i$