Trigonometry Examples

z = 2 i

$z = 2 i$

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

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

| 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 positive, the angle of the point on the complex plane is

\frac{π}{2}

$\frac{π}{2}$ .

θ = \frac{π}{2}

$θ = \frac{π}{2}$

Step 7

Substitute the values of

θ = \frac{π}{2}

$θ = \frac{π}{2}$ and

| z | = 2

$| z | = 2$ .

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

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

Step 8

Replace the right side of the equation with the trigonometric form.

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

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

z = 2 i

$z = 2 i$

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