Trigonometry Examples

${(2 - 2 i)}^{2}$

Step 1

Rewrite ${(2 - 2 i)}^{2}$ as $(2 - 2 i) (2 - 2 i)$ .

$(2 - 2 i) (2 - 2 i)$

Step 2

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

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Step 2.1

Apply the distributive property.

$2 (2 - 2 i) - 2 i (2 - 2 i)$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $2$ by $2$ .

$4 + 2 (- 2 i) - 2 i \cdot 2 - 2 i (- 2 i)$

Step 3.1.2

Multiply $- 2$ by $2$ .

$4 - 4 i - 2 i \cdot 2 - 2 i (- 2 i)$

Step 3.1.3

Multiply $2$ by $- 2$ .

$4 - 4 i - 4 i - 2 i (- 2 i)$

Step 3.1.4

Multiply $- 2 i (- 2 i)$ .

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Step 3.1.4.1

Multiply $- 2$ by $- 2$ .

$4 - 4 i - 4 i + 4 i i$

Step 3.1.4.2

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

$4 - 4 i - 4 i + 4 (i^{1} i)$

Step 3.1.4.3

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

$4 - 4 i - 4 i + 4 (i^{1} i^{1})$

Step 3.1.4.4

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

$4 - 4 i - 4 i + 4 i^{1 + 1}$

Step 3.1.4.5

Add $1$ and $1$ .

$4 - 4 i - 4 i + 4 i^{2}$

Step 3.1.5

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

$4 - 4 i - 4 i + 4 \cdot - 1$

Step 3.1.6

Multiply $4$ by $- 1$ .

$4 - 4 i - 4 i - 4$

Step 3.2

Subtract $4$ from $4$ .

$0 - 4 i - 4 i$

Step 3.3

Subtract $4 i$ from $0$ .

$- 4 i - 4 i$

Step 3.4

Subtract $4 i$ from $- 4 i$ .

$- 8 i$

Step 4

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 5

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 6

Substitute the actual values of $a = 0$ and $b = - 8$ .

$| z | = \sqrt{{(- 8)}^{2}}$

Step 7

Find $| z |$ .

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Step 7.1

Raise $- 8$ to the power of $2$ .

$| z | = \sqrt{64}$

Step 7.2

Rewrite $64$ as $8^{2}$ .

$| z | = \sqrt{8^{2}}$

Step 7.3

Pull terms out from under the radical, assuming positive real numbers.

$| z | = 8$

Step 8

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{- 8}{0})$

Step 9

Since the argument is undefined and $b$ is negative, the angle of the point on the complex plane is $\frac{3 π}{2}$ .

$θ = \frac{3 π}{2}$

Step 10

Substitute the values of $θ = \frac{3 π}{2}$ and $| z | = 8$ .

$8 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$