Trigonometry Examples

$\frac{1}{9 + 2 i}$

Step 1

Multiply the numerator and denominator of $\frac{1}{9 + 2 i}$ by the conjugate of $9 + 2 i$ to make the denominator real.

$\frac{1}{9 + 2 i} \cdot \frac{9 - 2 i}{9 - 2 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{1 (9 - 2 i)}{(9 + 2 i) (9 - 2 i)}$

Step 2.2

Multiply $9 - 2 i$ by $1$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(9 + 2 i) (9 - 2 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $9$ by $9$ .

$\frac{9 - 2 i}{81 + 9 (- 2 i) + 2 i \cdot 9 + 2 i (- 2 i)}$

Step 2.3.2.2

Multiply $- 2$ by $9$ .

$\frac{9 - 2 i}{81 - 18 i + 2 i \cdot 9 + 2 i (- 2 i)}$

Step 2.3.2.3

Multiply $9$ by $2$ .

$\frac{9 - 2 i}{81 - 18 i + 18 i + 2 i (- 2 i)}$

Step 2.3.2.4

Multiply $- 2$ by $2$ .

$\frac{9 - 2 i}{81 - 18 i + 18 i - 4 i i}$

Step 2.3.2.5

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

$\frac{9 - 2 i}{81 - 18 i + 18 i - 4 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{9 - 2 i}{81 - 18 i + 18 i - 4 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{9 - 2 i}{81 - 18 i + 18 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{9 - 2 i}{81 - 18 i + 18 i - 4 i^{2}}$

Step 2.3.2.9

Add $- 18 i$ and $18 i$ .

$\frac{9 - 2 i}{81 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add $81$ and $0$ .

$\frac{9 - 2 i}{81 - 4 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{9 - 2 i}{81 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{9 - 2 i}{81 + 4}$

Step 2.3.4

Add $81$ and $4$ .

$\frac{9 - 2 i}{85}$

Step 3

Split the fraction $\frac{9 - 2 i}{85}$ into two fractions.

$\frac{9}{85} + \frac{- 2 i}{85}$

Step 4

Move the negative in front of the fraction.

$\frac{9}{85} - \frac{2 i}{85}$