To express the complex number [tex]\(-9\)[/tex] in polar form, we need to determine its modulus and argument.
1. Modulus Calculation (r):
The modulus of a complex number [tex]\( z = a + bi \)[/tex] is given by [tex]\( |z| = \sqrt{a^2 + b^2} \)[/tex].
For the given number [tex]\(-9\)[/tex]:
[tex]\[
a = -9, \quad b = 0
\][/tex]
Hence,
[tex]\[
|z| = \sqrt{(-9)^2 + 0^2} = \sqrt{81} = 9
\][/tex]
2. Argument Calculation ([tex]\(\theta\)[/tex]):
The argument [tex]\(\theta\)[/tex] is the angle made with the positive real axis. It can be calculated using the atan2 function. For a purely real negative number:
[tex]\[
\theta = \pi + \frac{\pi}{2} = \frac{3\pi}{2}
\][/tex]
3. Polar Form Representation:
The polar form of a complex number is given by:
[tex]\[
z = r (\cos \theta + i \sin \theta)
\][/tex]
Substituting the calculated values [tex]\( r = 9 \)[/tex] and [tex]\( \theta = \frac{3\pi}{2} \)[/tex]:
[tex]\[
z = 9\left( \cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2} \right)
\][/tex]
Therefore, the polar form of [tex]\(-9\)[/tex] is:
[tex]\[
9\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)
\][/tex]
Among the given choices, this corresponds to:
\[
\boxed{9\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)}
\
