Express -3+1 in polar form.
Answers
Answer:
-3+1=3-1=2
Is it right?
Answer:
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Step-by-step explanation:
The polar form of a complex number is
R
e
i
θ
, where
R
is the number's modulus (its distance from
0
) and
θ
is the angle formed by the positive real axis and the number's vector on the complex plane.
We have a nice way of converting to polar coordinates by using Euler's formula:
e
i
θ
=
cos
(
θ
)
+
i
sin
(
θ
)
. Thus, if we can find and factor out
R
, we can find (theta) from the remaining number.
In this case, we will first find
2
+
i
in polar form, and then apply the power of
1
2
.
To find
R
, we find the number's modulus:
|
a
+
b
i
|
=
√
a
2
+
b
2
|
2
+
i
|
=
√
2
2
+
1
2
=
√
5
⇒
2
+
i
=
√
5
(
2
√
5
+
1
√
5
i
)
So, we have
cos
(
θ
)
=
2
√
5
and
sin
(
θ
)
=
1
√
5
As
arccos
(
2
√
5
)
=
arcsin
(
1
√
5
)
≈
0.4636
is not one of the "nice" angles, we'll leave it in that form. For ease of use, let's let
θ
0
=
arccos
(
2
√
5
)
and write that for the remainder of the problem.
Proceeding, we now have
2
+
i
=
√
5
(
cos
(
θ
0
)
+
i
sin
(
θ
0
)
)
By Euler's formula, this gives us
2
+
i
=
√
5
e
i
θ
0
Note that we can add
2
π
i
in any integer multiple without changing the value due to the cyclic nature of
sin
(
θ
)
and
cos
(
θ
)
. This will become relevant once we take the root.
2
+
i
=
√
5
e
i
(
θ
0
+
2
π
n
)
Finally, we take a power of
1
2
to get
√
2
+
i
=
(
√
5
e
i
θ
0
)
1
2
=
5
1
4
e
i
θ
0
+
2
π
n
2
=
5
1
4
e
i
(
θ
0
2
+
π
n
)