Trigonometry Examples

$(x + y i) (x - y i)$

Step 1

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

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Step 1.1

Apply the distributive property.

$x (x - y i) + y i (x - y i)$

Step 1.2

Apply the distributive property.

$x \cdot x + x (- y i) + y i (x - y i)$

Step 1.3

Apply the distributive property.

$x \cdot x + x (- y i) + y i x + y i (- y i)$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $x \cdot x + x (- y i) + y i x + y i (- y i)$ .

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Step 2.1.1

Reorder the factors in the terms $x (- y i)$ and $y i x$ .

$x \cdot x - i x y + i x y + y i (- y i)$

Step 2.1.2

Add $- i x y$ and $i x y$ .

$x \cdot x + 0 + y i (- y i)$

Step 2.1.3

Add $x \cdot x$ and $0$ .

$x \cdot x + y i (- y i)$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $x$ by $x$ .

$x^{2} + y i (- y i)$

Step 2.2.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.2.1

Move $y$ .

$x^{2} + y \cdot y i (- 1 i)$

Step 2.2.2.2

Multiply $y$ by $y$ .

$x^{2} + y^{2} i (- 1 i)$

Step 2.2.3

Rewrite $- 1 i$ as $- i$ .

$x^{2} + y^{2} i (- i)$

Step 2.2.4

Multiply $y^{2} i (- i)$ .

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Step 2.2.4.1

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

$x^{2} + y^{2} (- (i^{1} i))$

Step 2.2.4.2

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

$x^{2} + y^{2} (- (i^{1} i^{1}))$

Step 2.2.4.3

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

$x^{2} + y^{2} (- i^{1 + 1})$

Step 2.2.4.4

Add $1$ and $1$ .

$x^{2} + y^{2} (- i^{2})$

Step 2.2.5

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

$x^{2} + y^{2} (- - 1)$

Step 2.2.6

Multiply $- 1$ by $- 1$ .

$x^{2} + y^{2} \cdot 1$

Step 2.2.7

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

$x^{2} + y^{2}$