To express the complex number [tex]\(\sin 120^\circ - i \cos 120^\circ\)[/tex] in polar form, we follow the steps below:
### Step 1: Determine the Cartesian Form of the Complex Number
Given the complex number:
[tex]\[
z = \sin 120^\circ - i \cos 120^\circ
\][/tex]
First, we find the values for [tex]\(\sin 120^\circ\)[/tex] and [tex]\(\cos 120^\circ\)[/tex].
- [tex]\(\sin 120^\circ\)[/tex]:
[tex]\[
\sin 120^\circ = \frac{\sqrt{3}}{2} \approx 0.8660254037844387
\][/tex]
- [tex]\(\cos 120^\circ\)[/tex]:
[tex]\[
\cos 120^\circ = -\frac{1}{2} \approx -0.4999999999999998
\][/tex]
Substituting these values into the complex number:
[tex]\[
z = 0.8660254037844387 - i(-0.4999999999999998)
\][/tex]
Simplifying, we get:
[tex]\[
z = 0.8660254037844387 + 0.4999999999999998i
\][/tex]
### Step 2: Calculate the Magnitude [tex]\( r \)[/tex]
The magnitude [tex]\( r \)[/tex] of a complex number [tex]\( a + bi \)[/tex] is given by:
[tex]\[
r = \sqrt{a^2 + b^2}
\][/tex]
For the complex number [tex]\( z = 0.8660254037844387 + 0.4999999999999998i \)[/tex]:
[tex]\[
r = \sqrt{(0.8660254037844387)^2 + (0.4999999999999998)^2}
\][/tex]
Substituting the values:
[tex]\[
r = \sqrt{0.8660254037844387^2 + 0.4999999999999998^2}
\][/tex]
[tex]\[
r = \sqrt{0.7499999999999998 + 0.25} = \sqrt{0.9999999999999998} \approx 1.0
\][/tex]
### Step 3: Calculate the Argument [tex]\( \theta \)[/tex]
The argument [tex]\( \theta \)[/tex] (or phase angle) of a complex number [tex]\( a + bi \)[/tex] is given by:
[tex]\[
\theta = \tan^{-1}\left(\frac{b}{a}\right)
\][/tex]
Given [tex]\( a = 0.8660254037844387 \)[/tex] and [tex]\( b = 0.4999999999999998 \)[/tex]:
[tex]\[
\theta = \tan^{-1}\left(\frac{0.4999999999999998}{0.8660254037844387}\right)
\][/tex]
[tex]\[
\theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right)
\][/tex]
By calculating the above:
[tex]\[
\theta \approx 29.999999999999993^\circ
\][/tex]
### Step 4: Express the Complex Number in Polar Form
The polar form of the complex number is written as:
[tex]\[
z = r(\cos \theta + i \sin \theta)
\][/tex]
From the calculations:
[tex]\[
r = 1.0
\][/tex]
[tex]\[
\theta \approx 29.999999999999993^\circ
\][/tex]
So, the polar form of the complex number [tex]\(\sin 120^\circ - i \cos 120^\circ\)[/tex] is:
[tex]\[
z = 1.0 \left(\cos 29.999999999999993^\circ + i \sin 29.999999999999993^\circ \right)
\][/tex]
Therefore, the complex number [tex]\(\sin 120^\circ - i \cos 120^\circ\)[/tex] expressed in polar form is:
[tex]\[
1.0 \angle 29.999999999999993^\circ
\][/tex]
