Parts of complex numbers

z = 4- 2i

Step-by-step explanation:

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

)

)

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

|

z

|

=

√

a

2

+

b

2

where

z

=

a

+

b

i

Substitute the actual values of

a

=

4

and

b

=

−

2

.

|

z

|

=

√

(

−

2

)

2

+

4

2

Find

|

z

|

.

|

z

|

=

2

√

5

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

θ

=

arctan

(

−

2

4

)

Since inverse tangent of

−

2

4

produces an angle in the fourth quadrant, the value of the angle is

−

0.4636476

.

θ

=

−

0.4636476

Substitute the values of

θ

=

−

0.4636476

and

|

z

|

=

2

√

5

.

2

√

5

(

cos

(

−

0.4636476

)

+

i

sin

(

−

0.4636476

)

)

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

z

=

2

√

5

(

cos

(

−

0.4636476

)

+

i

sin

(

−

0.4636476

)

)

5

step-by-step explanation:

3 3 6, because in every triangle the longer side is always shorter than the sum of the other two