Unit Test Unit Test Active 21 22 23 24 25 For the complex number $z=\frac{5 \sqrt{3}}{4}-\frac{5}{4} i$, what is the polar form? $z=\frac{5}{2}\left(\cos \left(\frac{\pi}{6}\right)+i \sin \left(\frac{\pi}{6}\right)\right.$ $z=\frac{5}{4}\left(\cos \left(\frac{\pi}{6}\right)+i \sin \left(\frac{\pi}{6}\right)\right.$ $z=\frac{5}{2}\left(\cos \left(\frac{11 \pi}{6}\right)+i \sin \left(\frac{11 \pi}{6}\right)\right.$ $z=\frac{5}{4}\left(\cos \left(\frac{11 \pi}{6}\right)+i \sin \left(\frac{11 \pi}{6}\right)\right.$ Mark this and return Save and Exit

Step 1 ：The question is asking for the polar form of a complex number. The polar form of a complex number is given by \(r(\cos(\theta) + i\sin(\theta))\) where \(r\) is the magnitude of the complex number and \(\theta\) is the angle it makes with the real axis.

Step 2 ：The magnitude of a complex number \(a + bi\) is given by \(\sqrt{a^2 + b^2}\) and the angle is given by \(\arctan(\frac{b}{a})\) if \(a > 0\) and \(\arctan(\frac{b}{a}) + \pi\) if \(a < 0\).

Step 3 ：In this case, the complex number is \(\frac{5 \sqrt{3}}{4}-\frac{5}{4} i\). So, \(a = \frac{5 \sqrt{3}}{4}\) and \(b = -\frac{5}{4}\).

Step 4 ：Let's calculate the magnitude and the angle.

Step 5 ：The magnitude of the complex number is 2.5 and the angle is -0.5235987755982989 radians. However, the angle is negative which means the complex number is in the fourth quadrant. To convert this to a positive angle, we add \(2\pi\) to the angle.

Step 6 ：Also, the options for the question are in terms of \(\pi\). So, we need to express the angle in terms of \(\pi\).

Step 7 ：The magnitude of the complex number is 2.5 and the angle is \(\frac{11\pi}{6}\).

Step 8 ：Looking at the options, the correct polar form of the complex number is \(z=\frac{5}{2}\left(\cos \left(\frac{11 \pi}{6}\right)+i \sin \left(\frac{11 \pi}{6}\right)\right.\)

Step 9 ：\(\boxed{z=\frac{5}{2}\left(\cos \left(\frac{11 \pi}{6}\right)+i \sin \left(\frac{11 \pi}{6}\right)\right.}\)