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
)
)