Trigonometry Examples

$\frac{- 37 - 5 i}{- 3 - 5 i}$

Step 1

Multiply the numerator and denominator of $\frac{- 37 - 5 i}{- 3 - 5 i}$ by the conjugate of $- 3 - 5 i$ to make the denominator real.

$\frac{- 37 - 5 i}{- 3 - 5 i} \cdot \frac{- 3 + 5 i}{- 3 + 5 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(- 37 - 5 i) (- 3 + 5 i)}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(- 37 - 5 i) (- 3 + 5 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$\frac{- 37 (- 3 + 5 i) - 5 i (- 3 + 5 i)}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{- 37 \cdot - 3 - 37 (5 i) - 5 i (- 3 + 5 i)}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{- 37 \cdot - 3 - 37 (5 i) - 5 i \cdot - 3 - 5 i (5 i)}{(- 3 - 5 i) (- 3 + 5 i)}$

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 $- 37$ by $- 3$ .

$\frac{111 - 37 (5 i) - 5 i \cdot - 3 - 5 i (5 i)}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.2.1.2

Multiply $5$ by $- 37$ .

$\frac{111 - 185 i - 5 i \cdot - 3 - 5 i (5 i)}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.2.1.3

Multiply $- 3$ by $- 5$ .

$\frac{111 - 185 i + 15 i - 5 i (5 i)}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.2.1.4

Multiply $- 5 i (5 i)$ .

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Step 2.2.2.1.4.1

Multiply $5$ by $- 5$ .

$\frac{111 - 185 i + 15 i - 25 i i}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.2.1.4.2

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

$\frac{111 - 185 i + 15 i - 25 (i^{1} i)}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.2.1.4.3

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

$\frac{111 - 185 i + 15 i - 25 (i^{1} i^{1})}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.2.1.4.4

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

$\frac{111 - 185 i + 15 i - 25 i^{1 + 1}}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.2.1.4.5

Add $1$ and $1$ .

$\frac{111 - 185 i + 15 i - 25 i^{2}}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.2.1.5

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

$\frac{111 - 185 i + 15 i - 25 \cdot - 1}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.2.1.6

Multiply $- 25$ by $- 1$ .

$\frac{111 - 185 i + 15 i + 25}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.2.2

Add $111$ and $25$ .

$\frac{136 - 185 i + 15 i}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.2.2.3

Add $- 185 i$ and $15 i$ .

$\frac{136 - 170 i}{(- 3 - 5 i) (- 3 + 5 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(- 3 - 5 i) (- 3 + 5 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{136 - 170 i}{- 3 (- 3 + 5 i) - 5 i (- 3 + 5 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{136 - 170 i}{- 3 \cdot - 3 - 3 (5 i) - 5 i (- 3 + 5 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{136 - 170 i}{- 3 \cdot - 3 - 3 (5 i) - 5 i \cdot - 3 - 5 i (5 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $- 3$ by $- 3$ .

$\frac{136 - 170 i}{9 - 3 (5 i) - 5 i \cdot - 3 - 5 i (5 i)}$

Step 2.3.2.2

Multiply $5$ by $- 3$ .

$\frac{136 - 170 i}{9 - 15 i - 5 i \cdot - 3 - 5 i (5 i)}$

Step 2.3.2.3

Multiply $- 3$ by $- 5$ .

$\frac{136 - 170 i}{9 - 15 i + 15 i - 5 i (5 i)}$

Step 2.3.2.4

Multiply $5$ by $- 5$ .

$\frac{136 - 170 i}{9 - 15 i + 15 i - 25 i i}$

Step 2.3.2.5

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

$\frac{136 - 170 i}{9 - 15 i + 15 i - 25 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{136 - 170 i}{9 - 15 i + 15 i - 25 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{136 - 170 i}{9 - 15 i + 15 i - 25 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{136 - 170 i}{9 - 15 i + 15 i - 25 i^{2}}$

Step 2.3.2.9

Add $- 15 i$ and $15 i$ .

$\frac{136 - 170 i}{9 + 0 - 25 i^{2}}$

Step 2.3.2.10

Add $9$ and $0$ .

$\frac{136 - 170 i}{9 - 25 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{136 - 170 i}{9 - 25 \cdot - 1}$

Step 2.3.3.2

Multiply $- 25$ by $- 1$ .

$\frac{136 - 170 i}{9 + 25}$

Step 2.3.4

Add $9$ and $25$ .

$\frac{136 - 170 i}{34}$

Step 3

Cancel the common factor of $136 - 170 i$ and $34$ .

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Step 3.1

Factor $34$ out of $136$ .

$\frac{34 \cdot 4 - 170 i}{34}$

Step 3.2

Factor $34$ out of $- 170 i$ .

$\frac{34 \cdot 4 + 34 (- 5 i)}{34}$

Step 3.3

Factor $34$ out of $34 (4) + 34 (- 5 i)$ .

$\frac{34 (4 - 5 i)}{34}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $34$ out of $34$ .

$\frac{34 (4 - 5 i)}{34 (1)}$

Step 3.4.2

Cancel the common factor.

$\frac{34 (4 - 5 i)}{34 \cdot 1}$

Step 3.4.3

Rewrite the expression.

$\frac{4 - 5 i}{1}$

Step 3.4.4

Divide $4 - 5 i$ by $1$ .

$4 - 5 i$