Trigonometry Examples

- 2 i

$- 2 i$

Step 1

This is the trigonometric form of a complex number where

| z |

$| z |$ is the modulus and

θ

$θ$ is the angle created on the complex plane.

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

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

Step 2

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

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

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

z = a + b i

$z = a + b i$

Step 3

Substitute the actual values of

a = 0

$a = 0$ and

b = - 2

$b = - 2$ .

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

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

Step 4

Find

| z |

$| z |$ .

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Step 4.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

| z | = \sqrt{4}

$| z | = \sqrt{4}$

Step 4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 4.3

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

| z | = 2

$| z | = 2$

| z | = 2

$| z | = 2$

Step 5

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

θ = \arctan (\frac{- 2}{0})

$θ = \arctan (\frac{- 2}{0})$

Step 6

Since the argument is undefined and

b

$b$ is negative, the angle of the point on the complex plane is

\frac{3 π}{2}

$\frac{3 π}{2}$ .

θ = \frac{3 π}{2}

$θ = \frac{3 π}{2}$

Step 7

Substitute the values of

θ = \frac{3 π}{2}

$θ = \frac{3 π}{2}$ and

| z | = 2

$| z | = 2$ .

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

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