Trigonometry Examples

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

Step 1

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

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $- i i$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

Add $1$ and $1$ .

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

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

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

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

Step 2.2.3.2

Multiply $- 1$ by $- 1$ .

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

Step 2.2.4

Reorder $2 i$ and $1$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Add $1$ and $1$ .

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

Step 2.3.5

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

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

Step 3

Move the negative one from the denominator of $\frac{1 + 2 i}{- 1}$ .

$- 1 \cdot (1 + 2 i)$

Step 4

Rewrite $- 1 \cdot (1 + 2 i)$ as $- (1 + 2 i)$ .

$- (1 + 2 i)$

Step 5

Apply the distributive property.

$- 1 \cdot 1 - (2 i)$

Step 6

Multiply $- 1$ by $1$ .

$- 1 - (2 i)$

Step 7

Multiply $2$ by $- 1$ .

$- 1 - 2 i$