Trigonometry Examples

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

Step 1

Multiply the numerator and denominator of $\frac{1}{0.5 - 0.25 \cdot i}$ by the conjugate of $0.5 - 0.25 \cdot i$ to make the denominator real.

$\frac{1}{0.5 - 0.25 \cdot i} \cdot \frac{0.5 + 0.25 i}{0.5 + 0.25 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{1 (0.5 + 0.25 i)}{(0.5 - 0.25 \cdot i) (0.5 + 0.25 i)}$

Step 2.2

Multiply $0.5 + 0.25 i$ by $1$ .

$\frac{0.5 + 0.25 i}{(0.5 - 0.25 \cdot i) (0.5 + 0.25 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(0.5 - 0.25 \cdot i) (0.5 + 0.25 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{0.5 + 0.25 i}{0.5 (0.5 + 0.25 i) - 0.25 \cdot i (0.5 + 0.25 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{0.5 + 0.25 i}{0.5 \cdot 0.5 + 0.5 (0.25 i) - 0.25 \cdot i (0.5 + 0.25 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{0.5 + 0.25 i}{0.5 \cdot 0.5 + 0.5 (0.25 i) - 0.25 \cdot i \cdot 0.5 - 0.25 \cdot i (0.25 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $0.5$ by $0.5$ .

$\frac{0.5 + 0.25 i}{0.25 + 0.5 (0.25 i) - 0.25 \cdot i \cdot 0.5 - 0.25 \cdot i (0.25 i)}$

Step 2.3.2.2

Multiply $0.25$ by $0.5$ .

$\frac{0.5 + 0.25 i}{0.25 + 0.125 i - 0.25 \cdot i \cdot 0.5 - 0.25 \cdot i (0.25 i)}$

Step 2.3.2.3

Multiply $0.5$ by $- 0.25$ .

$\frac{0.5 + 0.25 i}{0.25 + 0.125 i - 0.125 \cdot i - 0.25 \cdot i (0.25 i)}$

Step 2.3.2.4

Multiply $0.25$ by $- 0.25$ .

$\frac{0.5 + 0.25 i}{0.25 + 0.125 i - 0.125 \cdot i - 0.0625 \cdot i i}$

Step 2.3.2.5

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

$\frac{0.5 + 0.25 i}{0.25 + 0.125 i - 0.125 \cdot i - 0.0625 \cdot (i^{1} i)}$

Step 2.3.2.6

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

$\frac{0.5 + 0.25 i}{0.25 + 0.125 i - 0.125 \cdot i - 0.0625 \cdot (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{0.5 + 0.25 i}{0.25 + 0.125 i - 0.125 \cdot i - 0.0625 \cdot i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{0.5 + 0.25 i}{0.25 + 0.125 i - 0.125 \cdot i - 0.0625 \cdot i^{2}}$

Step 2.3.2.9

Subtract $0.125 \cdot i$ from $0.125 i$ .

$\frac{0.5 + 0.25 i}{0.25 + 0 i - 0.0625 \cdot i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Multiply $0$ by $i$ .

$\frac{0.5 + 0.25 i}{0.25 + 0 - 0.0625 \cdot i^{2}}$

Step 2.3.3.2

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

$\frac{0.5 + 0.25 i}{0.25 + 0 - 0.0625 \cdot - 1}$

Step 2.3.3.3

Multiply $- 0.0625$ by $- 1$ .

$\frac{0.5 + 0.25 i}{0.25 + 0 + 0.0625}$

Step 2.3.4

Add $0.25$ and $0$ .

$\frac{0.5 + 0.25 i}{0.25 + 0.0625}$

Step 2.3.5

Add $0.25$ and $0.0625$ .

$\frac{0.5 + 0.25 i}{0.3125}$

Step 3

Rewrite $0.5$ as $1 (0.5)$ .

$\frac{1 (0.5) + 0.25 i}{0.3125}$

Step 4

Factor $1$ out of $0.25 i$ .

$\frac{1 (0.5) + 1 (0.25 i)}{0.3125}$

Step 5

Factor $1$ out of $1 (0.5) + 1 (0.25 i)$ .

$\frac{1 (0.5 + 0.25 i)}{0.3125}$

Step 6

Factor $0.3125$ out of $0.3125$ .

$\frac{1 (0.5 + 0.25 i)}{0.3125 (1)}$

Step 7

Separate fractions.

$\frac{1}{0.3125} \cdot \frac{0.5 + 0.25 i}{1}$

Step 8

Divide $1$ by $0.3125$ .

$3.2 \frac{0.5 + 0.25 i}{1}$

Step 9

Divide $0.5 + 0.25 i$ by $1$ .

$3.2 (0.5 + 0.25 i)$

Step 10

Apply the distributive property.

$3.2 \cdot 0.5 + 3.2 (0.25 i)$

Step 11

Multiply $3.2$ by $0.5$ .

$1.6 + 3.2 (0.25 i)$

Step 12

Multiply $0.25$ by $3.2$ .

$1.6 + 0.8 i$